Sansa could never understand how two sisters, born only two years apart, could be so different. It would have been easier if Arya had been a bastard, like their half brother Jon. She even looked like Jon, with the long face and brown hair of the Starks, and nothing of their lady mother in her face or her coloring. And Jon's mother had been common, or so people whispered. Once, when she was littler, Sansa had even asked Mother if perhaps there hadn't been some mistake. Perhaps the grumkins had stolen her real sister. But Mother had only laughed and said no, Arya was her daughter and Sansa's trueborn sister, blood of their blood. Sansa could not think why Mother would want to lie about it, so she supposed it had to be true.
(AGoT, Sansa I, Chapter 15)
You know how GRRM likes foreshadowing? And irony? That passage about Arya being a trueborn Stark, who does not look like Sansa, is in Sansa's first POV.
If Arya is trueborn, what if Sansa is the bastard?
Catelyn tells us that she was a virgin when she married Ned
And when Brandon was murdered and father told me I must wed his brother, I did so gladly, though I never saw Ned's face until our wedding day. I gave my maidenhood to this solemn stranger and sent him off to his war and his king and the woman who bore him his bastard, because I always did my duty.
(ACoK, Catelyn VI, Chapter 45)
We know that Robb was born in 283 AC at Riverrun while Ned was fighting in Robert's Rebellion. After Ned was done fighting, he returned to Winterfell with the baby Jon Snow who was identified as Ned's bastard son. Catelyn must have soon arrived at Winterfell with baby Robb. With the bastard child Jon Snow in full view, the marriage was off to a rocky start.
I'm thinking Littlefinger snuck up to Winterfell at some point during these difficult early years. Maybe he pretended to be part of the construction crew for the sept Ned had built for Catelyn's use. Baelish had a fling with his longtime love, now Lady Stark. Sansa, Catelyn's second child, born in 286 AC, would be the daughter of Petyr Baelish. Sansa doesn't look like a Stark because she isn't a Stark; she's different from the trueborn daughter because she is a bastard; she's a Baelish.
This theory could help to explain some of Petyr's motives. If he really was behind the catspaw with the dragonbone and Valyrian steel dagger, he might want Bran out of the way so Sansa would be closer to inheriting Winterfell. He might want to laugh behind Ned's back as Ned discovers the history of Baratheon hair color and the mismatch with Cersei's children. Can Ned not see the hair color and features of his own daughter? Littlefinger might want Joffrey out of the way because he was threatening Sansa even after the betrothal was ended and because Joffrey's death created an opportunity to get Sansa out of the Red Keep without anyone seeing Littlefinger in connection with her escape.
I have to admit, some of my clues for this Sansa = Alayne Baelish theory work only because I have a long-held suspicion that Littlefinger is a hidden Targaryen of some kind -- could be a Brightflame, Blackfyre or Velaryon descendant, but he is surrounded by Targaryen hints and clues. I think he has ambitions toward the Iron Throne but not for himself: he is trying to position Sansa to be the ruler of Westeros.
So at the same time this Baelish, sneaking into Winterfell and impregnating Lady Stark, resembles Bael the Bard, sneaking into Winterfell to seduce a Stark daughter, we are also seeing Jenny of Oldstones and the Prince of Dragonflies in the Petyr / Catelyn love affair. Catelyn recalls role-playing where she was Jenny with flowers in her hair and Petyr was the Prince of Dragonflies. This is even more meaningful as a parallel if Littlefinger really is some kind of Targaryen with a claim to the throne.
A couple of aspects of Sansa's escape from the Red Keep also struck me as potentially meaningful. First, she is aided and accompanied by Ser Dontos, whose family sigil is three crowns. He is the closest surviving relative of the Darklyns, who were the most loyal members of the kingsguard over many generations of Targaryen rule. Does his return to knighthood as he helps Sansa escape mean that he becomes something of a kingsguard for her? The two of them run down the serpentine steps in the Red Keep on their way out of the castle. Many non-Targs go up or down these steps, but they are compared to the serpentine neck of Dany's dragon, on which she escapes from Daznak's Pit. GRRM may want us to compare Sansa on the steps with Dany climbing onto Drogon's back.
Second, during their "flight" from the Red Keep, Sansa and Dontos run through the long gallery, a room filled with suits of scaled armor, the scales being associated with dragon scales:
They continued down the serpentine and across a small sunken courtyard. Ser Dontos shoved open a heavy door and lit a taper. They were inside a long gallery. Along the walls stood empty suits of armor, dark and dusty, their helms crested with rows of scales that continued down their backs. As they hurried past, the taper's light made the shadows of each scale stretch and twist. The hollow knights are turning into dragons, she thought.
One more stair took them to an oaken door banded with iron. "Be strong now, my Jonquil, you are almost there." When Dontos lifted the bar and pulled open the door, Sansa felt a cold breeze on her face. She passed through twelve feet of wall, and then she was outside the castle, standing at the top of the cliff. Below was the river, above the sky, and one was as black as the other.
ASoS, Sansa V
The imagery of the flame bringing the knights to life as dragons is definitely a Targaryen symbol: a dragon hatching. I couldn't figure out why this would be included in Sansa's story but it falls into place if she is a child of Petyr Baelish and he is a Targaryen.