THE FOLLOWING THEORY MENTION EVENTS FROM AGOT TO ADWD, SPOILER ALERT.
How I came to this theory is a curious thing and it wasn't until the events of AFFC where we first read Cersei Lannister’s POV chapters that I noticed a failure. And it is that we see that Lyanna Stark occupies part of her thoughts, because she considers that if things had not turned bitter between her father Tywin and King Aerys II, she would have married Rhaegar, he would not have fallen in love with Lyanna and the rebellion of Robert Baratheon would not have happened. The problem is that she never mentions her aunt to Sansa Stark, even though their stories are so "similar". They both fell in love with the prince who turned out to be cruel, Sansa with Joffrey and Lyanna with Rhaegar, because in the eyes of the Seven Kingdoms Rhaegar kidnapped and raped Lyanna. I am not GRRM to tell him how to write his books but I found it very strange that Cersei never release a comment (with which she said she wanted to help Sansa to realize about the truth of the world) about her aunt Lyanna's destiny. Now all this may seem only an omission without importance but I believe that Cersei Lannister hides this information because the last thing she needs at the moment is people to start wondering about Rhaegar and Lyanna. Why? Because those who paid more attention to those events would know the truth that Jon Snow, that he is son of Rhaegar and Lyanna. Well so far I have only told you a hunch and something that most of us accept (R + L = J) now come my proof.
The conversation between Cersei and Jaime Lannister in Winterfell. Here they refer to the incest between them and the truth about the kinship of their children, when they mention Lysa Arryn and if she has evidence they refer to this matter. This is the only way that Ned Stark could find out the truth since he has been all those years away from King’s Landing. But this dialogue is out of that context.
“He betrayed one already, or have you forgotten?” the woman said. “Oh, I don’t deny he’s loyal to Robert, that’s obvious. What happens when Robert dies and Joff takes the throne? And the sooner that comes to pass, the safer we’ll all be. My husband grows more restless every day. Having Stark beside him will only make him worse. He’s still in love with the sister, the insipid little dead sixteen-year-old. How long till he decides to put me aside for some new Lyanna?”
Here Cersei Lannister suspects that Eddard Stark plans something against her and her children, the mention of Lyanna and Joffrey as king points to Jon Snow's possibility of revealing his identity and claiming his rights when Robert Baratheon dies. If the truth of Joffrey, Tommen and Myrcella were discovered and King Robert assassinated them, he would put one of his bastards as heir, which would leave the path to Jon Snow practically free to claim the crown at Robert Baratheon’s death.
The story that Edric Dayne tells to Arya Stark makes no sense.
— How do you know about Jon?
— He is my milk brother.
— Brother? — Arya did not understand. — But you’re from Dorne. How could you and Jon be blood?
— Milk brothers. Not blood. My lady mother had no milk when I was little, so Wylla had to nurse me.
Arya was lost. —Who’s Wylla?
— Jon Snow’s mother. He never told you? She’s served us for years and years. Since before I was born.
— My father was Ser Arthur’s elder brother. Lady Ashara was my aunt. I never knew her, though. She threw herself into the sea from atop the Palestone Sword before I was born.
— Why would she do that?” said Arya, startled.
Ned looked wary. Maybe he was afraid that she was going to throw something at him.
— Your lord father never spoke of her? — He said. — The Lady Ashara Dayne, of Starfall?
— No. Did he know her?
— Before Robert was king. She met your father and his brothers at Harrenhal, during the year of the false spring.
— Oh. — Arya did not know what else to say. — Why did she jump in the sea, though?
— Her heart was broken.
— Did someone break it?
He hesitated. — Perhaps it’s not my place . . .
— Tell me.
He looked at her uncomfortably. — My aunt Allyria says Lady Ashara and your father fell in love at Harrenhal.
There is great incongruity here. Most people in Westeros affirm that Jon Snow is the son of Ashara Dayne, but the numbers make it impossible, the Harrenhal Tournament was in 281 AC, even if it were in the last days of the year, a son of a fortuitous encounter between Eddard Stark and Ashara Dayne would be born in 282 AC which would leave Jon about one year old when Eddard returned with him to Winterfell. That would have been clearly noticed by Catelyn Tully. The only way would be that during the year between Harrenhal and the “kidnapping” of Lyanna, Eddard and Ashara would have seen each other again. Now if Jon Snow is the son of a woman Eddard met in the war, it would have to be practically from the beginning to give enough time for him to be born. Well, Edric Dayne states that Jon Snow's mother is Wylla, a servant at the service of the Dayne house, but in that case, Wylla was the entire war in Starfall because I doubt she would accompany Ashara since she was a lady of Princess Elia Martell. Therefore the numbers fail again because if Eddard knew Wylla until the end of the war, the issue of the time necessary for Jon's birth is still standing. And it is worth mentioning that Catelyn claims that Ned returned with Jon and a nurse. Knowing Eddard Stark he would have allowed Wylla "Jon's mother" to go to Winterfell, obviously if the Dayne allowed her to leave. Why Edric Dayne states that Wylla is Jon's mother but then mentions the love between Eddard and Ashara? It can be assumed that Eddard's infidelity was what broke Ashara's heart, but again, why Wylla would stay in Starfall? I doubt that Ashara or his brothers liked her after that and she still breastfed Edric several years later. Well here comes another character in the theory, Allyria Dayne, she has told Edric these things, and I think she has been lying, it is curious that Lord Dayne is death and we do not know anything about Edric's mother, I think Allyria planned the death of her brother and sister to become Lady of Starfall.
Cersei Lannister's way of referring to Jon Snow when she discusses with the Private Council about the loyalty of the Night's Watch. First, it is Cersei Lannister who brings the issue of Jon Snow to the discussion and in the end she wants to take care of the matter on her own.
— Another problem has arisen on the Wall, however. The brothers of the Night’s Watch have taken leave of their wits and chosen Ned Stark’s bastard son to be their Lord Commander.
— It will need to be done carefully, to be sure. Leave the rest to me, my lords.
Here we must emphasize the differences in how Cersei refers to Jon Snow in her thoughts and how when she speaks, while she mentions him, leaves no doubt that he is the son of Eddard Stark, but the two thoughts in this discussion are very strange.
— I glimpsed him once at Winterfell, — the queen said, —though the Starks did their best to hide him. He looks very like his father.
— Snow shares Lord Eddard’s taste for treason too, — she said. — The father would have handed the realm to Stannis. The son has given him lands and castles.
« Catelyn Tully was a mouse, or she would have smothered this Jon Snow in his cradle. Instead, she’s left the filthy task to me »
« If this bastard boy is truly his father’s son, he will not suspect a thing. Perhaps he will even thank me, before the blade slides between his ribs. »
When she says he will not suspect anything, she can refer to Littlefinger's betrayal of Eddard. But what does she mean when she says that he can even thank her when he is murdered? Does she mean that she free him from the weight of his name by killing him in anonymity? Second, she says that Catelyn must have killed him when he was a baby. Why? He does not pose a threat to anyone. Even though Cersei had threatened Mya Stone and possibly when Robert died it was she who sent to kill Gendry and Robert's other baseborn, actually did nothing for years. Jon is different because of the threat his identity.
The reaction of Cersei Lannister upon learning of the attack on his daughter Myrcella by Gerold Dayne. It can be interpreted that at this point Cersei discovers why Tyrion sent Myrcella to Dorne.
Ser Kevan gave her another scowl. — Myrcella was attacked by a Dornish knight named Gerold Dayne. She’s alive, but hurt. He slashed her face open, she … I’m sorry … she lost an ear.
— An ear.” Cersei stared at him, aghast. She was just a child, my precious princess. She was so pretty, too. — He cut off her ear. And Prince Doran and his Dornish knights, where were they? They could not defend one little girl? Where was Arys Oakheart?
— Slain, defending her. Dayne cut him down, it’s said.
The Sword of the Morning had been a Dayne, the queen recalled, but he was long dead. Who was this Ser Gerold and why would he wish to harm her daughter? She could not make any sense of this, unless…
But why Gerold Dayne attacked Myrcella remains a mystery. In the chapter of Arianne Martell when she plans to crown Myrcella, Gerold Dayne draws his sword when he mentions that they should kill Myrcella. If we continue with the theory, Gerold found out what happened to his relatives Dayne and confronted Allyria, I don't know if he killed her or not, but I think he took Dawn from Starfall and with it he plans to take revenge. Cersei has realized that someone have discovered the murder of the Dayne family, mainly Ashara, and that someone could know the truth about Jon Snow, but does not trust anyone at that time so she decide to tell his uncle Kevan Lannister anything .
— The lion’s dead. Who knows which cub the lioness prefers?
— The one in her own den. — Ser Gerold drew his sword. It glimmered in the starlight, sharp as lies. — This is how you start a war. Not with a crown of gold, but with a blade of steel.
Ser Gerold went to one knee. The moonlight shone in his dark eyes as he studied the child coolly.
Lastly, there are two mentions of Jaime Lannister. First, when he throws Bran Stark from the tower his quote is so strange coming from him, I think it was Ashara Dayne's last words before he killed her. Second, the similarities between the White Sword tower and the Palestone Sword tower and how Jaime feels being there.
The man looked over at the woman. — The things I do for love, — he said with loathing. He gave Bran a shove.
Those rooms were spare as well, though spacious; and they were above the outer walls, which meant he would have a view of the sea.
« I will like that », he thought. « The view, and all the rest. »
As pale as the room, Jaime sat by the book in his Kingsguard whites, waiting for his Sworn Brothers.
Finally all this leads me to the conversation between Cersei Lannister and Eddard Stark in the forest of Old Gods.
"Do you?" The queen watched his face, wary as a cat. "Is that why you called me here, Lord Stark? To pose me riddles? Or is it your intent to seize me, as your wife seized my brother?"
"If you truly believed that, you would never have come." Ned touched her cheek gently. "Has he done this bufare?"
Here Cersei suspects that Eddard is trying a plan but she does not know it. She attends the meeting trying to find out what his intentions are. The fact that she goes with simple clothes is to avoid being recognized along with him.
"You know what I must do."
"Must?" She put her hand on his good leg, just above the knee. "A true man does what he will, not what he must." Her fingers brushed lightly against his thigh, the gentlest of promises. "The realm needs a strong Hand. Joff will not come of age for years. No one wants war again, least of all me." Her hand touched his face, his hair. "If friends can turn to enemies, enemies can become friends. Your wife is a thousand leagues away, and my brother has fled. Be kind to me, Ned. I swear to you, you shall never regret it."
Which war Cersei refers to? The fact that her children turned out to be the result of incest would not cause a great war. Possibly Tywin Lannister would rise for the murder of his grandchildren but who else would follow him, he is not very popular among the other kingdoms. It would happen as with Balon Greyjoy’s rebellion, it would be crushed immediately, I think she means that if Eddard plans to raise Jon Snow as Rhaegar’s heir that would cause a war between the houses that were loyal to the Targaryen and those that was not. She proposed him to leave that plan and accept Joffrey Baratheon as king and let everything takes its course.
"I shall wear that as a badge of honor," Ned said dryly.
"Honor," she spat. "How dare you play the noble lord with me! What do you take me for? You've a bastard of your own, I've seen him. Who was the mother, I wonder? Some Dornish peasant you raped while her holdfast burned? A whore? Or was it the grieving sister, the Lady Ashara? She threw herself into the sea, I'm told. Why was that? For the brother you slew, or the child you stole? Tell me, my honorable Lord Eddard, how are you any different from Robert, or me, or Jaime?"
Why would Cersei mention Jon Snow? And why would she wonder about his mother? From the beginning she made it clear that she did not want to play riddles. She wants to know if Eddard Stark really plans to reveal the identity of Jon Snow and overthrow his son Joffrey. In the end she questions him that what he is doing would be an equally dishonorable act.
The queen stood. "And what of my wrath, Lord Stark?" she asked softly. Her eyes searched his face. "You should have taken the realm for yourself. It was there for the taking. Jaime told me how you found him on the Iron Throne the day King's Landing fell, and made him yield it up. That was your moment. All you needed to do was climb those steps, and sit. Such a sad mistake."
"I have made more mistakes than you can possibly imagine," Ned said, "but that was not one of them."
"Oh, but it was, my lord," Cersei insisted. "When you play the game of thrones, you win or you die. There is no middle ground."
Finally Cersei thinks that Eddard will continue with his plan to use Robert to get rid of the Lannister and then take the crown through Jon Snow She suggests him that if that was his intention, he should have taken the realm for himself when he had the opportunity. That his whole plan will fail.
My theory is that Cersei Lannister began to suspect, when rumors of Eddard Stark's visit to Ashara Dayne in Starfall and that he left with a child, arrived at King's Landing. At this point Cersei should know who Eddard Stark was as much for what his brother Jaime told her as for what Robert Baratheon could have told her, in her eyes Eddard Stark would be a man who would never stain his honor, so the boy had to come from another side. The idea must have jumped into his mind immediately, the time that Robert Baratheon's rebellion lasted was long enough for Lyanna Stark to have given birth to a son, whether it was a consensual relation or not. That child put at risk everything she had achieved and also the legitimacy of the throne of her future children. But she could not act without evidence and still did not know Eddard Stark's intentions to take the child. She would not believe that he would dare to betray his best friend Robert Baratheon raising the child under a false identity and revealing it when he has the age to start a rebellion against his King. She sent his brother Jaime Lannister to Starfall to find out what he could, because she knew that in the North he would not be well received and would not obtain information. He interrogated Ashara Dayne and I don't know if he could get the truth out of it or only part of it, but in the end he killed her by shoving her from the tower. Her younger sister Allyria Dayne helped him cover up the murder, with the aim of becoming Lady de Starfall. Jaime Lannister returned with his sister Cersei and told her what happened, at this point I do not know what could have happened but I am sure that Cersei discovered the truth, although I doubt that Jaime Lannister knew what it was all about or did not care. In the events of a Song of Ice and Fire, Cersei continues to suspect that Eddard Stark has a plan for Jon Snow. After his conversation with him in the forest of Old Gods she is more convinced. Her plan to send him to the Night's Watch will serve to that Eddard cannot do anything but without provoking a war with the North. Finally and although the execution of Eddard Stark was not in her plans, she knows that the secret has died with him so she leaves the matter aside. And it is not until Jon Snow is appointed commander of the Night's Watch and that Gerold Dayne attacks Myrcella, that this issue becomes important again.