One of the mysteries of the period between the Harrenhal Tourney in late 281 AC and the execution of Lord Rickard and Brandon Stark in 282 AC, is Brandon's decision to look for Rhaegar at the Red Keep after he heard about Lyanna's abduction, when we know that Rhaegar wasn't there, and that Rhaegar actually lived on Dragonstone, not at the Red Keep.
There are a number of convoluted theories about how Brandon heard about Rhaegar's abduction of Lyanna, what Brandon was told about Rhaegar's abduction of Lyanna, and how some mastermind (usually a fifteen year old Petyr Baelish) intentionally manipulated Brandon into rushing to the Red Keep to die.
But I think there is a simple and plausible explanation for why Brandon rode to the Red Keep believing Rhaegar was to be found there: Rhaegar himself intentionally misled witnesses to his abduction of Lyanna to believe that he was going to be taking her to the Red Keep.
And I think we are provided with a basis for that explanation in the very first book of A Song of Ice and Fire, in A Game of Thrones, when Catelyn intentionally misleads witnesses to her abduction of Tyrion to believe that she was going to be taking him to Winterfell, when her real intention was to bring him to her sister at the Eyrie in the Vale.
"In the name of King Robert and the good lords you serve, I call upon you to seize him and help me return him to Winterfell to await the king's justice."
- AGOT: Catelyn V
We are taking him back to Winterfell," she said, and Tyrion thought, Well, perhaps … By then he'd had a moment to glance over the room and get a better idea of the situation. He was not altogether displeased by what he saw. Oh, the Stark woman had been clever, no doubt of it. Force them to make a public affirmation of the oaths sworn her father by the lords they served, and then call on them for succor, and her a woman, yes, that was sweet. Yet her success was not as complete as she might have liked. There were close to fifty in the common room by his rough count. Catelyn Stark's plea had roused a bare dozen; the others looked confused, or frightened, or sullen. Only two of the Freys had stirred, Tyrion noted, and they'd sat back down quick enough when their captain failed to move. He might have smiled if he'd dared.
"Winterfell it is, then," he said instead. That was a long ride, as he could well attest, having just ridden it the other way. So many things could happen along the way. "My father will wonder what has become of me," he added, catching the eye of the swordsman who'd offered to yield up his room. "He'll pay a handsome reward to any man who brings him word of what happened here today." Lord Tywin would do no such thing, of course, but Tyrion would make up for it if he won free.
Ser Rodrik glanced at his lady, his look worried, as well it might be. "His men come with him," the old knight announced. "And we'll thank the rest of you to stay quiet about what you've seen here."
It was all Tyrion could do not to laugh. Quiet? The old fool. Unless he took the whole inn, the word would begin to spread the instant they were gone. The freerider with the gold coin in his pocket would fly to Casterly Rock like an arrow. If not him, then someone else. Yoren would carry the story south. That fool singer might make a lay of it. The Freys would report back to their lord, and the gods only knew what he might do. Lord Walder Frey might be sworn to Riverrun, but he was a cautious man who had lived a long time by making certain he was always on the winning side. At the very least he would send his birds winging south to King's Landing, and he might well dare more than that.
Catelyn Stark wasted no time. "We must ride at once. We'll want fresh mounts, and provisions for the road. You men, know that you have the eternal gratitude of House Stark. If any of you choose to help us guard our captives and get them safe to Winterfell, I promise you shall be well rewarded." That was all it took; the fools came rushing forward. Tyrion studied their faces; they would indeed be well rewarded, he vowed to himself, but perhaps not quite as they imagined.
They set out through the rain at a hard gallop, and before long Tyrion's thighs were cramped and aching and his butt throbbed with pain. Even when they were safely away from the inn, and Catelyn Stark slowed them to a trot, it was a miserable pounding journey over rough ground, made worse by his blindness. Every twist and turn put him in danger of falling off his horse. The hood muffled sound, so he could not make out what was being said around him, and the rain soaked through the cloth and made it cling to his face, until even breathing was a struggle. The rope chafed his wrists raw and seemed to grow tighter as the night wore on. I was about to settle down to a warm fire and a roast fowl, and that wretched singer had to open his mouth, he thought mournfully. The wretched singer had come along with them. "There is a great song to be made from this, and I'm the one to make it," he told Catelyn Stark when he announced his intention of riding with them to see how the "splendid adventure" turned out. Tyrion wondered whether the boy would think the adventure quite so splendid once the Lannister riders caught up with them.
The rain had finally stopped and dawn light was seeping through the wet cloth over his eyes when Catelyn Stark gave the command to dismount. Rough hands pulled him down from his horse, untied his wrists, and yanked the hood off his head. When he saw the narrow stony road, the foothills rising high and wild all around them, and the jagged snowcapped peaks on the distant horizon, all the hope went out of him in a rush. "This is the high road," he gasped, looking at Lady Stark with accusation. "The eastern road. You said we were riding for Winterfell!"
Catelyn Stark favored him with the faintest of smiles. "Often and loudly," she agreed. "No doubt your friends will ride that way when they come after us. I wish them good speed."
Even now, long days later, the memory filled him with a bitter rage. All his life Tyrion had prided himself on his cunning, the only gift the gods had seen fit to give him, and yet this seven-times-damned she-wolf Catelyn Stark had outwitted him at every turn. The knowledge was more galling than the bare fact of his abduction.
- AGOT: Tyrion IV
As a result, witnesses spread the tale that Catelyn was bringing Tyrion to Winterfell when she was actually bringing him to the Eyrie in the Vale. In Rhaegar's case, witnesses would have spread the tale that he was bringing Lyanna to King's Landing when he was actually bringing her elsewhere.
Of course, in Catelyn's case, we know exactly why she abducted Tyrion, while we can only speculate why Rhaegar abducted Lyanna. Was it to fulfill prophecy through her? Was it out of love or desire for her? Was it to save her from her marriage to Robert? Was it to make her a captive on behalf of his father Aerys?
Personally, I think Rhaegar abducted Lyanna to protect her against Aerys.
Lyanna was the Knight of the Laughing Tree
Aerys was furious about the KOTLT, believing him to be Jaime, and sent Rhaegar to seek him
Rhaegar discovered that Lyanna was the KOTLT
The KOTLT vanished without Aerys learning his identity
Aerys was certain that someone close to him had warned the "traitor" known as the KOTLT
Aerys and his closest small councilors (Chelsted and Staunton) believed Rhaegar entered the tourney as part of his plot to depose Aerys and become king
Rhaegar crowned Lyanna the Queen of Love and Beauty because he discovered she was the KOTLT
Aerys and his closest small councilors (Chelsted and Staunton) took the crowning to mean that Rhaegar was looking to bring the Starks into his plot to depose Aerys and become king
Aerys and his small councilors eventually learned that Lyanna was in fact the KOTLT, and connected both the KOTLT and QOLAB scenes as part of the same plot
Aerys commanded that Lyanna be arrested and brought to King's Landing
Therefore, I think that Rhaegar was either commanded to bring Lyanna to Aerys himself (as Jaime would later be commanded to bring Tywin's head to Aerys), or learned of Aerys's intentions to have her arrested, and set out to get to her before those who Aerys commanded to bring her to him did.
Admittedly, I do not have all the details worked out, and though I think this is a potential explanation for why Rhaegar abducted Lyanna, I think it is more speculative than the idea that Rhaegar was responsible for the disinformation that he was bringing Lyanna to the Red Keep.