To understand this theory, there are a few things to keep in mind about Jon Snow. For most of the show’s run, he was thought to be the bastard offspring of Ned Stark and a mysterious woman named Wylla. She’s not mentioned much in the show, but in the novels the show is based on, Edric Dayne tells Arya that Wylla was a servant at the Dornish castle of Starfall.
In season six, though, it was revealed that Jon wasn’t Ned’s son after all. Indeed, a flashback showed Ned doing battle with Ser Arthur Dayne outside the Tower of Joy. After defeating Dayne, Ned ran into the tower to find his sister in the throes of death, having given birth to a son – Jon. She asks Ned to protect him from Robert Baratheon, with the implication being that Jon is Prince Rhaegar Targaryen’s offspring.
At the end of season seven, this was expanded on further. It was Bran who first figured that Jon had Targaryen blood, but it was Sam Tarly who filled in the last piece of the puzzle. Indeed, a book he’d found in the Citadel revealed that Rhaegar and Lyanna had actually wed in secret. This means that the bastard of Winterfell isn’t a bastard at all. No, his real name is Prince Aegon Targaryen.