Spoiler/theory about Jon Snow (mainly hinted at, or, imo, blatently telling you, in the first book)
Lyanna Stark made Ned promise her something before her death. Howland Reed being the only other person alive to have heard those words (father of Bran's new mentor dude in the series
) It is widely thought that Rhaegar Targaryen fathered a son with Lyanna Stark and that that son is Jon Snow. Making him not a bastard at all (Rhaegar potentially took her as a wife, as he had more than one anyways if my memory serves me) and also Daenerys's Nephew. The book series is called 'A song of Ice and Fire' also, with Jon representing Ice and Daenerys Fire.