A big part of this theory stems from a prophecy in the books, which was omitted from the show. In the books, when Dany is in the House of the Undying in Qarth, she has a vision of her brother Rhaegar, holding his infant son, Aegon, and he says, “There needs to be one more, the dragon has three heads.” Some fans have interpreted this quote as meaning that there will be three dragonriders, one for each of Dany’s dragons. Assuming that Dany rides Drogon, who would get Viserion and Rhaegal?
The theory posits that since Bran has tremendous warging abilities, which will presumably only grow stronger under the influence of the Three-Eyed Raven, he will warg into one of the dragons when the time comes, likely to fight against the White Walkers. This would make Bran a dragonrider in a sense; since he can’t physically ride one, controlling it mentally is just as good (if not better).
More evidence for this theory comes from something we have seen in the show that the books corroborate. When Bran meets the Three-Eyed Raven, he tells Bran that he’ll never walk again, but he will fly. If he’s warging into a dragon, that sure sounds like he’ll be flying to us!
Whether Bran controls a dragon or not, this theory assumes that Dany eventually makes it to Westeros, which might just be the craziest fan prediction of all!