In the books when Ned dies his ghost appears to all his children. He leaves a message for Jon with Arya. When Arya asks him why he can't deliver it himself he says he can only appear to his children or something similar. Basically Ned != Jon's dad. The only other Starks alive at that point were Lyanna and Benjen. So if he's truly of Stark blood, which Ned says he is, chances are he's Lyanna's son, and Rhaegar was hitting that all the damn time, which was what started the rebellion in the first place.