Many readers believe that Jon is not the son of Eddard Stark. Instead, he is the son of Prince Rhaegar Targaryen and Eddard's sister Lyanna. Rhaegar and Lyanna disappeared together to the Tower of Joy early in Robert's Rebellion.
There, it's believed,
Rhaegar leaves a pregnant Lyanna to defend his family's dynasty.
At the end of Robert's Rebellion, about one year later
, Eddard and his companions find three of the Kingsguard at the Tower of Joy: Lord Commander Ser Gerold Hightower, Ser Oswell Whent, and Ser Arthur Dayne. The reasons for their presence and the ensuing fight are unknown, but defending the unborn son of the Heir Apparent
would be a good reason to have been posted. The only known survivors of the fight were Eddard himself and Howland Reed. Eddard recalls his sister dying "in a bed of blood,"
where he made her an unknown promise
just before she died.
Also, while Robb, Sansa, Bran and Rickon are said to have Tully features (hair color, eyes), Jon and Arya
are said to be closer in appearance (which had made Sansa believe Arya was also a bastard like Jon, until her mother put Sansa's theory down). Arya is said to resemble Lyanna.
Further evidence to the truth of this theory in the eighth Eddard chapter of A Game of Thrones, in which Ned contemplates the significance of King Robert's bastards. As he muses, Ned's thoughts drift to Jon Snow, a logical segue, but also to his sister Lyanna, the promise he made her, and to Rhaegar Targaryen,
implying some tacit link between the three individuals.
Daenerys Targaryen's visions in the House of the Undying include an image of Rhaegar with his newborn son Aegon, proclaiming that "there must be a third"
because "the dragon has three heads".
Given that, according to Jon Connington, Rhaegar's wife Elia was believed infertile after two difficult pregnancies, and that Aegon the Conqueror himself had two wives, it is logical for Rhaegar to have attempted to fulfill the prophecy by having a third child with another woman.
Another of the visions Daenerys sees is the image of a blue winter rose growing from a chink in a wall of ice, filling the air with sweetness.
Lyanna was noted by Ned to be fond of winter roses, and he associates them with her death. Jon, who could be the product of Lyanna, is currently at the Wall.
If Rhaegar secretly married Lyanna, it would ironically mean that Jon is not a bastard despite his life having been defined by his believed bastard status to a great extent, and that he is the heir to the Iron Throne after Aegon VI Targaryen (although this last point is rendered moot so long as he belongs to the Night's Watch).
With Eddard beheaded by King Joffrey, Howland Reed is the only living person who knows the nature of Lyanna's death and what she made her brother promise. Source