All the Game of Thrones Fan Theories You Absolutely Need to Know
These are pretty fun. I especially like the theory about Tyrion being part Targaryen.
Jon's parentage seems likely to most readers – so likely, in fact, that some people believe it's too obvious to be true. Other fans have theorized that Jon is genuinely Ned's child, possibly with some random woman, or possibly with Ashara Dayne, whom Ned was rumored to love. The question is whether Ned's honor is great enough to have precluded him from fathering a bastard — certainly Catelyn and Ned's friends were shocked at the idea Ned had cheated –- and as shown in GoT, Ned is honorable to a fault. So it seems unlikely, although if he did lapse, his shame might be the reason he's so quiet about Jon's parentage. But if his mother is Ashara Dayne or some random wench, that doesn't really explain why he wouldn't at least tell Jon.
Certainly, George R.R. Martin loves his head games, but the idea that Rhaegar and Lyanna are Jon';s true parents makes the most narrative sense. And it seems unlikely that Martin would be hinting at the mystery so much only to reveal Jon is just Ned's regular ol' bastard.