10Septon Maynard's Journal
The simplest, but by far the least likely way that Jon could gain the throne for himself would be the acceptance of his claim - courtesy of Septon Maynard's journal. This was the book that Gilly and Sam found at the Citadel, a diary in which the High Septon
recorded the annulment of Rhaegar's marriage to Elia Martell, and his re-marriage to Lyanna Stark.
If this was one of the books that Sam stole when he left the citadel, Jon could use it - along with Bran/The Three-Eyed Raven's visions, to prove that he is the rightful heir and gain the support of the Lords of Westeros. However, even if Jon can prove his claim, it's incredibly unlikely that the other players in the Game of Thrones would simply roll over and accept it.