After having reread through the first book I have come to the conclusion that Lyanna Stark and Rhaegar Targyren must have had an elopement of some sort and/or a child.
Here is the largest piece of evidence, short (as possible) and sweet (and rather difficult to ignore afterwords).
In a Game Of Thrones, Chapter 39, Eddard X,
we begin with the scene at the Tower Of Joy.
Eddard is dreaming of it from the past. He confronts three men, "in white cloaks, and Lyanna in her bed ablood." He is accompanied by six northern men he knew well but sees as shadows in the dream. The three men in white cloaks are Sir Arthur Dayne, Oswell Whent, and Lord Commander Gerold Hightower. They have a brief discussion, in which Gerold Hightower explains that "they made vow's " draw swords and then Eddard and his shadows descend on the three. Ned hears his sister screaming his name, as he is awakened he keeps repeating, "I promise, Lya, I promise."
Now to put it in retrospect.
The key here is the conversation they had in the tower and what it reveals, and equally important, who the men are and their motives. Fortunately GRRM has given us just the right amount of clever subtly while at the same time not giving it away without a lot of thought.
All three white cloaked men are member's of the King's Guard. An elite force of 7, sworn to protect the king. Why is The Lord Commander of the King's Guard neither with the prince when he died, nor with the king when he died, nor with the young Viserys and Dannerys, as they have just escaped recently. Why is nearly half the King's Guard been dispatched to guard Lyanna Stark, as the Seven Kingdoms tare apart from civil war? Why is Rhagar's Best friend, Sir Aruthur Dane not with him at the most pivotal battle in "Roberts Rebellion"? To defened Lyanna Stark from her own brother? Lyanna Stark who is commonly known as merely a rape and kidnap victim of Rhaegar Targaryen? Unless the opposite is true.
Then I saw the answer in what Gerold Hightower says to Eddard, right before Gerold is killed, "we swore a vow." Surely they couldn't believe that their Prince's last will was to keep Lyanna Stark who is clearly suffering from some mortal wound, from her brother, who is clearly not interested in doing anything but saving the life of his sister.
The only possible thing left in my mind, is that Rhagar eloped with Lyanna, impregnated her, and left her where he thought safe and away, in a tower in the mountains of Dawn. A tower called the Tower of Joy. Where she could safely give birth to his third child. (Lots of prophecy speculation I wont go into)
Put that in and it all makes sense. It fits into why the Lord Commander of the Kings Guard Gerold HIghtower explains they made a vow, his last words before death as far as we know. For the only surviving child of Rhagar would indeed be heir to the Iron Throne and therefore King.That is why they did not surrender to Ned, because the King inside Lyanna was in what they saw as danger.
Lyanna makes Ned swear that he must never reveal the identity of the child. From here we can only make two possiblities, either the child died with Lyanna or Eddard took that child in and claimed it as a bastard, thus explaining John snow's true parents.
I also find it interesting how calm Ned remains with Cercei when she defends her incest with the fact that Ned had a bastard and how she thinks he got the bastard. Before this Ned had treated all inquires with frank hostility and suddenly with Cercei he seems to be amused by how she brought it up. Does this sound like a man ashamed? He replies that the first difference between them is that "I do not kill children."
We also have never seen Eddard be sexual in an offensive manner. Many who are exposed by this folly, yet lord Eddard never is. He seems to have a very healthy relationship with his wife, even given that he has boldly taken in a bastard and treated him like one of his own.
Eddard Stark has proven to disgrace his honor, only to save children. He lied in front of a whole host in Kings Landing to save his daughters, it seems his only folly is lying for noble causes, that protect children.
This is not to say that GRRM could write any thing he wants, could even change his mind tomorrow. Mostly likely though, given what small pieces he has given us, we must conclude that the most likely parents of Jon Snow are Prince Rhaegar and Lyanna Stark. Probably after they got married, and assuming, as that would turn out, John couldnt be more true born, how ironic.