I'm actually now thoroughly convinced that Jon Snow is a Targaryen and also son of Ned's sister, Lyanna.
In the books, apparently (I haven't read them), the whole reason the Starks/Baratheons go to war vs the Targaryens is because Rhaegar kidnapped Lyanna Stark. Ned's brother went to save her, they locked him up. Ned's father went, they locked him up. When they demanded trial by combat, the Targaryens instead cut both their heads off. Right there, we see that it's completely possible that Rhaegar Targaryen impregnated Ned's sister.
From
http://awoiaf.westeros.org/index.php/Rhaegar_Targaryen
"The circumstances of Rhaegar's taking of Lyanna, and of her reaction to it, remain unknown. The Starks, Brandon Stark in particular, saw it as an abduction, and Robert Baratheon imagined that Rhaegar was raping Lyanna while he kept her prisoner. However, the Targaryen tradition sees Rhaegar acting out of true love for Lyanna. By the time Lyanna was found by her brother,Eddard, she was being kept in the Tower of Joy in Dorne where, on Rhaegar's command, she was guarded by three knights of the Kingsguard. Lyanna died soon after Eddard's arrival and his defeat of the knights. She made her brother give a promise, the content of which remains unknown, as Eddard apparently never shared Lyanna's secret with anyone."
I believe that it is around the time of Ned's return from the rebellion and defeat of the Targaryens that he brought home Jon Snow to Winterfell, to protect his sister's child under the guise of him being his bastard.
In the show, Ned was always kind of strange about who was Jon's mother, and he wouldn't even tell his own wife. Also, if you recall in Season 1, Robert Baratheon casually mentions the one wh0re who Ned alleged was "crazy in bed" or something of the sort (I believe it was to imply that it was Jon Snow's mother). In that scene, Ned gets pretty awkward. I know Ned was a noble and honorable guy, so cheating for him doesn't seem likely. It's much more believable that he made that story up to cover for his sister's bastard child. Mind you, the Lannisters killed every Targaryen they could get their hands on, and Jon Snow would have been dead as a baby if this information was leaked.
For this reason, I think he is too important to be permanently dead, particularly with Melisandre in Castle Black (and even Davos, who has seen her magic at work). If true, he literally has the blood of fire and ice [Targaryen/Stark] and that is in the name of the books.
Also, the tales of Azor Ahai that Melisandre tells can match up to Jon Snow and how Azor Ahai is supposed to be resurrected. I've spent WAY too much time thinking about this, but google Azor Ahai and it really seems like this is supposed to be Jon Snow, resurrected. BOOM.
