Since book one in the ASOIAF series, "A Game of Thrones," Martin has dropped several clues pointing to Jon Snow's lineage, and all signs point to him not being Ned Stark's bastard son.
An exchange between Stark and his wife Catelyn, who despises Snow and believes that her husband cheated on her and fathered him despite him never saying so, holds one clue.
"Never ask me about Jon," he said cold as ice. "He is my blood, and that is all you need know," reads a passage from Thrones.
The standout theory is that Ned's sister Lyanna, who was killed during king Robert Baratheon's rebellion to rid the Iron Throne of the Targaryens, is actually Jon Snow's mother, something Stark never reveals to Jon. This makes it possible that Snow's real father is Prince Rhaegar Targaryen, who "abducted" Lyanna when she was betrothed to Baratheon, setting off the war that ended with the surviving Targaryens -- Rhaegar's brother Viserys and sister Daenerys -- in exile across the Narrow Sea.
Through flashbacks, Ned reveals that his sister made him promise her something as she lay dying, but it isn't revealed what. All signs point to her making Ned promise to keep her son's secret safe and to raise him, so if Jon Snow is indeed Lyanna and Raegar's son, that makes him a Targaryen by blood.
Unfortunately, Snow, who becomes the Lord Commander of the Night's Watch in "A Dance With Dragons," is apparently killed by his brothers in the Night's Watch, which is where the series leavs off (and why everyone who's read it is dying for Martin to hurry up with "The Winds of Winter"!)
Several other signs and visions point to a possible resurrection, much like the resurrection of stepmother Catelyn Stark, who was brought back to life by the Red Priest Thoros of Myr.
The Red Woman, Melisandre, is also a believer in the same religion as Thoros, and she also possesses supernatural powers. She also basically lives with Snow at Castle Black and had a vision that perhaps he, and not Stannis Baratheon as she previously believed, is the true second coming of Azor Ahai, a hero who delivered the world from darkness with his magical flaming sword Lightbringer.
According to A Wiki of Ice and Fire, there are a handful of theories involving the second coming of Azor Ahai, though Snow's seems to stand out more. He has a dream that he fights off the Others with a burning sword in his hand while atop the Wall, and Melisandre says that while looking into the flames for visions, "I pray for a glimpse of Azor Ahai, and R'hllor shows me only Snow."
So could it be that Jon Snow single-handedly defeats the White Walkers and keeps the world from falling into darkness, and then places his aunt -- Daenerys -- on the Iron Throne, restoring the Targaryen rule over the Seven Kingdoms of Westeros?