Okay, if you've read all the books you probably guessed or have heard that Jon is not Eddard Stark's bastard son. If you've only seen/read the first season/book, you're ready to hear it, since most of the relevant hints are there even if you miss them (as I did my first time). The prevailing theory is that Ned's sister Lyanna had an affair with Rhaegar Targaryen, entrusting her son to her brother's care before she died in childbirth. Ned claimed the child as his own, easily accounting for his Starkish appearance, to protect his identity (you know how it was for Targs under Robert's reign).
I'm only halfway sold on this. My main beef is that I don't want Jon to be a Targ, because I ship him with Dany and we're all sick of incest. (Except for the Jon/Robb shippers...where did those come from, anyway?) It also doesn't sit right with me that he's has both ice and fire blood when everything in his story so far has been symbolic of ice while Dany is all fire. If it all comes down to him instead of the two of them, it diminishes her importance and I don't like that even symbolically.
So, I think it's practically a sure thing that Jon is Lyanna's son, but I'm still looking for his father. So far I've got at least some evidence backing me up on it not being Rhaegar: there's a line in Got: Riding through the rainy night, Ned saw Jon Snow's face in front of him, so like a younger version of his own. If the gods frowned so on bastards, he thought dully, why did they fill men with such lusts? There's no reason for him to bemoan Jon's lust, so he must be thinking of his conception, his father. Not a page later, we get this: For the first time in years, he found himself remembering Rhaegar Targaryen. Years didn't pass between those two pages.
Most speculation about R+L=J (Rhaegar and Lyanna make Jon - ASOIAF fandom abbreviates lineage instead of pairings) involves the "tower of joy" where Rhaegar was keeping Lyanna and where she ultimately died while Ned was trying to rescue her. Rhaegar wasn't there; he was fighting Robert on the Trident. However, the tower was guarded by no less than three knights of the Kingsguard - that's half the ones who stayed faithful to the monarchy. The Kingsguard's duty is to protect the royal family, the king first, then the heirs, the queen, etc. I don't know the full order of priority, but whatever it is, they must have been at that tower for someone of royal blood. The only one that could possibly be was Lyanna's baby, but even that doesn't add up: he would have been the bastard son of the heir, nowhere near the top of the chain of succession. Why would he merit that kind of protection?
My theory is that the Kingsguard understood that their monarch was insane, and they also understood that Rhaegar was a wise, just man and the best hope of the kingdom, and their allegiance was with him. Rhaegar spent a lot of time reading and apparently picked up on a few important prophecies; what if one of them was about Lyanna's child being needed to save the world from whatever? He could have tried to steal her away and set up some kind of circumstances to fulfill his prophecy - evidently thwarted, but we don't know if he was right about everything. The Kingsguard could have believed him, though, or even just obeyed out of loyalty.
That still doesn't give me a father, but I've been paying close attention to the stories of the First Men and the children of the forest - maybe a wildling dad? That would keep up the ice theme and give us a nice stark contrast (heh) to the Targaryens.
Didn't mean to go on that long. I wonder if anyone will read this? In any case, I love Jon and here's why. This is actually a passage from near the beginning of CoK, a conversation between Lord Mormont and Jon about Robb being crowned.
"They will garb your brother Robb in silks, satins, and velvets of a hundred different colors, while you live and die in black ringmail. He will wed some beautiful princess and father sons on her. You'll have no wife, nor will you ever hold a child of your own blood in your arms. Robb will rule, you will serve. Men will call you a crow. Him they'll call Your Grace. Singers will praise every little thing he does, while your greatest deeds all go unsung. Tell me that none of this troubles you, Jon...and I'll name you a liar, and know I have the truth of it."
Jon drew himself up, taut as a bowstring. "And if it did trouble me, what might I do, bastard as I am?"
"What will you do?" Mormont asked. "Bastard as you are?"
"Be troubled," said Jon, "and keep my vows."
Seriously, how can anyone not love that kid? I hope that line is kept in the show. Even if it is, this is premature, but Jon, you have my heart so I made you this: