The most popular, most prevailing and all in all most likely fan theory is the one referred to as "L + R = J." Basically, Jon Snow's real mother is Ned's sister Lyanna Stark, and her father is Daenerys' older brother, Prince Rhaegar Targaryen. Since Robert Baratheon killed Rhaegar before he became king and would smash the shit out of any royal babies that would challenge his right to the throne, Lyanna made her brother pledge to keep Jon safe by claiming him as his own. That's what all the stuff about "Promise me, Ned" means. The theory has a lot of evidence, and fills in a lot of holes in the history of Jon Snow and Westeros at large.
Too bad it's wrong. There's a dark horse for Targaryenhood...
Good ol' Samwell Tarly. Sam the Slayer, as he's called by his comrades and lovers. He's about an unlikely a Targ as they get, but let's look at the details. People think that Daenerys' vision of "A blue flower growing from a chink in a wall of ice, filling the air with sweetness" implicates Jon Snow, but it could just as easily point to his brother of the Night's Watch. Sam's dad hates him and refuses to make him the heir, which could be the result of his unmanly love of books (which coincidentally, Rhaegar Targaryen also shares), but also could be because he wasn't born a Tarly.
Sam also has a lot in common with Samwise Gamgee from The Lord of the Rings trilogy. Like Samwise, Samwell is a loyal sidekick to the hero (Jon Snow), sort of goofy and unassuming but eventually proves himself with heroic acts (stabbing white walkers). And really, even J.R.R. Tolkien thinks that Samwise was the real hero of the LotR trilogy. Why couldn't Samwell be the same for A Song of Ice and Fire?
It might sound like a stretch, but put yourself in George R. R. Martin's fancy orthopedic shoes: If you were setting someone up to be an unlikely hero, would you make him the prettyboy action star? Or would you choose the chubby kid that looks like a younger version of you?