With Season 6 right around the corner, and R+L=J becoming a mainstream fan theory, I think we've reached that point in the show where it's finally safe to answer this question. I urge you to stay with me till the end (I will answer the question eventually). What follows is speculative, and contains no confirmed spoilers (if you've watched till Season 5).
What the show has told us
Ned and Robert rebelled against the Mad King Aerys, who had murdered Ned's brother and father. Robert was betrothed to Ned's sister Lyanna, whom Aerys' son Rhaegar had earlier kidnapped and imprisoned.
Some time during Robert's rebellion, Ned bedded a yet-to-be-identified* woman, and Jon came tumbling after, curly hair et al.
Robert: She must have been a rare wench to make Lord Eddard Stark forget his honor. You never told me what she looked like.
Ned: Nor will I.
Robert: We were at war. None of us knew if we were going to go back home again. You're too hard on yourself. You always have been.
*Before this exchange, Ned does say that the woman he impregnated was called Wylla, but this is almost certainly something he said just to satiate Robert's curiosity, and not the truth.
The rebellion finally ended with Robert killing Rhaegar, Tywin sacking King's Landing, Jaime killing the Mad King, Ned branding Jaime Kingslayer, and Robert ascending the throne thanks to his part-Targaryen ancestors. In the midst of all this chaos, Lyanna died too, right in front of Ned's eyes.
After the war, Ned took his bastard son back with him to Winterfell. Many years later, Ned promised to talk to the bastard Jon about his mother when they meet next.
Jon: Is my mother alive? Does she know about me? Where I am, where I'm going? Does she care?
Ned: The next time we see each other, we'll talk about your mother. Huh. I promise.
Well, we know how that ended.
Now, there is almost unanimous consensus among fans that Jon Snow is in fact not Ned's bastard, but the son of Rhaegar Targaryen and Lyanna Stark -- a prophesied prince who would save all of humanity. This theory holds well because of what we know about the honorable Ned Stark (and the fact that the show has painstakingly established Jon as a pure hero from the start). Ned bedding someone other than his wife, be it a whore or a noblewoman, seems out of character regardless of whether he thought he was going to die in the war or not.
Furthermore, ranking high in the list of famous last words are Lyanna's.
Promise me, Ned!
This line serves as further vindication for fans of the R+L=J theory, as it has been widely interpreted as Lyanna imploring Ned to keep Jon's actual parentage a secret at all costs. In doing so, Lyanna demonstrates her love for Jon -- something that would be unlikely had Jon been born as a result of Rhaegar raping her. The R+L=J theory therefore goes on to ask whether Rhaegar did in fact "kidnap" Lyanna, or if the two instead fell in love, consensually eloped and birthed a child together. Once again, there is growing consensus that the latter is what actually happened.
Interlude: What we know about Thrones
If there's one thing we've learnt time and again from Thrones, it's that things are never as black and white as they seem. Take the example of Jaime Lannister. Introduced to us in the pilot as the incestuous Disney Villain who casually defenestrates little boys, Jaime after season 5 is ironically one of the good guys -- someone whom audiences have come to love, accept and sympathize with ().
The people's perception of Jaime started to change around the time that he told Brienne of what actually happened the day he stabbed the Mad King.
Senile and paranoid, Aerys would have burnt down the whole of King's Landing had Jaime not broken his oath to protect the king, and killed him.
The arriving Ned Stark however saw only a king stabbed in the back by one of his own Kingsguard, and immediately presumed that Jaime killed the king in order to take the throne for himself (and by proxy, for his father Tywin).
Key takeaway: If we had only Ned Stark's version of things to rely on, we would never know the truth about what happened with Jaime.
Key question: Can you think of any other character whose story we've heard only from Ned (and Robert's) point of view?
Aside: What we can infer about Ned
Throughout the 9 episodes that we saw him, Ned promulgated an obstinate brand of honorable living -- one that leaves no room for nuance. Be it the manner in which he left his wife behind to go be the King's Hand, or his being naive to the danger that he was putting his daughters in, honor and duty clearly outranked family in Ned's worldview.
What R+L=J means for Ned
Lyanna was betrothed to Robert, Ned's friend. So once that had been decided, it was her duty to marry him. In any conservative society, a couple running away to be together is nothing short of a scandal. Add to this a patriarchal world order, and the blame often falls on the girl for bringing dishonor to her house.
Furthermore, Ned's older brother Brandon, went to war against Aerys because he thought Lyanna had been kidnapped by Rhaegar. In the chain of events that ensued, the Mad King burnt Ned's father alive and caused Brandon's death.
Now, if Lyanna had eloped willingly with Rhaegar, she had not just besmirched the name of her house but also caused the deaths of her brother and father.
So tell me, faced with this evidence, wouldn't Ned judge her guilty? And doesn't the man who passes the sentence have to swing the sword?
Ned Stark: Honorable or Honor Killer?
Ned storms the Tower of Love in a bid to free Lyanna. Upon seeing Lyanna protecting a newborn child, he confronts her and learns the truth about her and Rhaegar. Perhaps she tells him of the prophecy. Perhaps she doesn't. Ned however has stopped listening. All he can think of his brother and father burning to death in the Red Keep.
His sister has sinned, and Ned must deliver justice. His hand shaking, Ned brandishes Ice and swings it at Lyanna. He cuts her, not clean, but enough to wound her fatally. She lies before him in a pool of blood. Lyanna is resigned to her fate. She knows that she can never justify her actions to her honorable brother. And with Rhaegar dead, what does she have to live for, except...
Not him, Ned! The child is innocent! Please Ned, promise me you won't hurt him! He's innocent! Promise me, Ned!
Ned has been known to show compassion to children in the TV show, going as far as offering Cersei a chance to flee the capital in order to save her kids from Robert's wrath. He therefore concedes that Lyanna's child is innocent and vows to protect him from having to suffer for his mother's sins (which is why he never reveals his true identity).
He takes the boy back home as his bastard son and spends the rest of his life believing what he did was right.
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