The character of Jon Snow has many mysteries surrounding him. This page tries to put forward some theories to give answers.
Jon Snow's Parentage
Potential Mothers with Eddard
In the knight of the Laughing Tree story that Meera Reed tells Bran Stark we are led to believe that Eddard had some feelings for Ashara Dayne, the sister of Ser Arthur Dayne, and that Ned's brother, Brandon Stark, asked if she would dance with him. Ser Barristan Selmy, another of Ashara's suitors, makes mention of how someone dishonoured her at Harrenhal and about how Ashara looked at a Stark. This has led many fans to believe that Ashara and Eddard were in a relationship at some point after the Tourney of Harrenhal.
It was a well known fact that Ashara was pregnant during Robert's Rebllion, but Barristan thinks that she has a stillborn girl.
When asked by his wife, Catelyn Stark, whether Ashara is the mother Ned tells Catelyn never to ask her that question again. Other believers of the theory include Queen Cersei Lannister.
Lord Edric Dayne, who goes by the name Ned, said that he was a milk brother to Jon Snow, claiming that he was Wylla's son. He says that they aren't blood brothers. Arya Stark denies his claims. Edric was nursed by Wylla as a babe.
Believers of this theory include King Robert Baratheon and Lord Edric Dayne.
The Fisherman's Daughter
This is a theory held on the Three Sisters. Lord Godric Borrell claims that, just after the beginning of Robert's Rebellion, a fisherman offered to row Ned out of the Vale of Arryn. The fisherman died but his daughter continued the rowing. Ned left her with a bag of silver and, according to Lord Godric, a bastard that she named after Jon Arryn.
This follows a similar vein to the theory about Ned and Ashara, saying that they both met at Harrenhal. We have further evidence, however, as Lady Barbrey Dustin tells us that Brandon was never afraid of taking what he wanted, and his relationship with her shows that he didn't mind dishonouring noble girls.
Potential Scenarios with Lyanna Stark as the Mother
The theory that Rhaegar and Lyanna had a children stems from the fact that Rhaegar kidnapped Lyanna, an act that indirectly caused Robert Baratheon, and whisked her away to the Tower of Joy, where Robert claims that he raped her. Rhaegar then leaves Lyanna to protect House Targaryen, but assigns Arthur Dayne, Gerold Hightower and Oswell Whent to protect her. They are knights of the Kingsguard, and it is speculated that they were left behind to guard the son of the Heir Apparent.
When Eddard finds Lyanna at the end of Robert's Rebellion he says that she is dying on a "bed of blood", which is possibly used to describe a bed used during a complicated birth. Lyanna then has Ned make a mysterious promise, possibly to pretend that Jon Snow is his own child.
When Daenerys Targaryen visits the House of the Undying she sees two visions that could lend themselves to this theory. The first is of her brother, Prince Rhaegar Targaryen, holding his second son and saying that a third was necessary, as the dragon must have three heads. Jon Connington tells us that Elia Martell was likely infertile, so if Rhaegar wanted a third child then he would have to look elsewhere. The second is of blue winter roses stuck in a chink in a wall of ice. Lyanna is associated with blue roses countless times, and Jon is currently at The Wall.
Jon Snow's Death
We have had exposition in the story for followers of R'hllor bring back the seemingly dead already. Thoros of Myr repeatedly brings back Beric Dondarrion, and the Lightning Lord himself goes on to revive Catelyn Stark as the vengeful Lady Stoneheart. Melisandre, another follower of the Red God, is currently at the Wall, and it is possible that she will be responsible for bringing Jon Snow back from the dead.
Becoming a Wight
The dead at the Wall and beyond it have a habit of being resurrected by the Others and made a part of their army of the undead. It is possible that this fate may happen to Jon, as predicted by Bran having a vision of him going incredibly pale. It remains to be seen whether Jon would end up a servant of the Others or a wight more like Coldhands, however.
Another possible solution to Jon's death problem is that he may have warged into Ghost, his direwolf. We see in the Varamyr Sixskins chapter that wargs can enter the minds of their animals upon death, and Jon's last thoughts are of Ghost. Melisandre also has a vision of Jon's face turning into a wolf's and then back into a man's, foreshadowing his fate.