There is no evidence that I have seen for the idea that Jon Snow is the bastard son of King Robert Baratheon and Lyanna, who was Eddard Stark’s sister. In fact, if we are to believe what George R.R. Martin says in the appendices to his books, Jon Snow cannot be their son. Instead, Snow is the bastard son of Eddard Stark and a woman whose identity is never revealed in the books.
In each of the books of this series there is an appendix that lists the names of various members of the important houses in the books. These appendices make it clear that Jon Snow is the son of Eddard Stark. For example, the appendix to A Game of Thrones has a listing of Eddard Stark’s relations. Among them is “his bastard son, Jon Snow, a boy of fourteen.” This makes it clear that Snow is Eddard’s son, not his nephew.
Another way to think of this is to think about how much Robert Baratheon still cherishes the memory of Lyanna. He is still in love with her years after her death. If they had had a bastard son, it is hard to imagine that Robert would not have kept the child with him.
All in all, then, there is no reason that I know of to think that Jon Snow is the son of Robert and Lyanna. Instead, he is the son of Eddard Stark and a woman whose identity is not disclosed to us.