Thett/ bossy Tournament R1 (ASOIAF) R + L = J is most likely the correct theory.

Seeing as we agree on the point of there being a child, my job is to counter my opponent's argument that it is Jon. His entire argument about Ned's reaction to the assasination plot applies to Daenerys moreso than it does to Jon; if Daenerys is actually the daughter of his sister, instead of being loosely related towards her, then the episode is more traumatic. As I have pointed out, the fact that Ned's haunting feelings of betrayal coincide with this confrontation of Robert are also explained much more clearly if Daenerys is the child. Jon is safe at the Wall, and the dreams intensify as Ned is in King's landing trying to convince Robert to abandon his plans.Consider this:'Yet even as he said the words, he remembered that chill morning on the barrowlands, and Robert's talk of sending hired knives after the Targaryen princess. He remembered Rhaegar's infant son, the red ruin of his skull, and the way the king had turned away, as he had turned away in Darry's audience hall not so long ago. He could still hear Sansa pleading, as Lyanna had pleaded once.'As Ned struggles with Robert's brutality, he mentally links the murder of Rhaegar's son, Lyanna's pleading, and Robert's plan to kill Daenerys. Also, there are signs that Jon is, in fact, not the son of Ned or Rhaegar. These come at two relatively early points.During one, Ned thinks to himself that 'for the first time in years, he found himself remembering Rhaegar Targaryen.' It is highly unlikely that, had Ned been raising Rhaegar's child, he would forget about him for years at a time. But if he had helped spirit that child across the narrow sea? That is much more likely.Then, there was the conversation with Catelyn.'They whispered of Ser Arthur Dayne, the Sword of the Morning, deadliest of the seven knights of Aerys’s Kingsguard, and of how their young lord had slain him in single combat. And they told how afterward Ned had carried Ser Arthur’s sword back to the beautiful young sister who awaited him in a castle called Starfall on the shores of the Summer Sea. The Lady Ashara Dayne, tall and fair, with haunting violet eyes. It had taken her a fortnight to marshal her courage, but finally, in bed one night, Catelyn had asked her husband the truth of it, asked him to his face.That was the only time in all their years that Ned had ever frightened her. 'Never ask me about Jon,' he said, cold as ice. 'He is my blood, and that is all you need to know. And now I will learn where you heard that name, my lady.' She had pledged to obey; she told him; and from that day on, the whispering had stopped, and Ashara Dayne’s name was never heard in Winterfell again.'This is incredibly revealing, because Catelyn never asks about Jon. She asks about rumors of an affair with Ashara Dayne, and Ned confirms by his response that there was a connection between Ashara Dayne and Jon. And this segues into my exploration of Jon's ancestry: A + B = JWhat do we know about these players? Well, we know that Ashara Dayne 'looked to Stark' and was dishonored during the Tourney at Harrenhal, as related by Barristan Selmy:'Even after all these years, Ser Barristan could still recall Ashara’s smile, the sound of her laughter. He had only to close his eyes to see her, with her long dark hair tumbling about her shoulders and those haunting purple eyes. Daenerys has the same eyes. Sometimes when the queen looked at him, he felt as if he were looking at Ashara’s daughter ...But Ashara’s daughter had been stillborn, and his fair lady had thrown herself from a tower soon after, mad with grief for the child she had lost, and perhaps for the man who had dishonored her at Harrenhal as well. She died never knowing that Ser Barristan had loved her. How could she? He was a knight of the Kingsguard, sworn to celibacy. No good could have come from telling her his feelings. No good came from silence either. If I had unhorsed Rhaegar and crowned Ashara queen of love and beauty, might she have looked to me instead of Stark?He would never know. But of all his failures, none haunted Barristan Selmy so much as that.'Ashara Dayne looked to Stark. This meant either Ned or Brandon, and considering Ned's honor, and Brandon's reputation as a womanizer, it is most likely the latter. Especially considering the fact that Howland Reed shared Ned's tent during the Tourney. So, what we know is that a Stark, most likely Brandon, deflowered Lady Ashara Dayne. That she became pregnant and that the child was stillborn. And that she killed herself after Ned Stark returned her family sword.We hear from Edric Dayne, who was nicknamed Ned after Eddard Stark, that his wetnurse Wylla was the mother of Jon Snow. Edric relates that he admired Ned and wanted to speak to him at the Hand's Tourney, but didn't know what to say. Now, if the stories are true (Ned deflowered and disgraced Ashara and killed her brother) it makes no sense for House Starfall to admire Ned, and to bring up the heir to their house with any sort of sense of respect for him. Unless Ned did something very important for House Starfall, something which would also explain Lady Dayne's suicide. And, if my theory is correct, he did.If Lady Ashara had a son by the deceased Brandon Stark who lived, removing him from Starfall and taking him on as his own bastard would save Starfall a LOT of trouble (titles pass through the female line in Dorne) while also simplifying matters in Winterfall. This is doubly true if Ashara and Brandon were secretly married, either on the Isle of Faces during the Harrenhal Tourney or later on. A male heir born from that union would be both heir to Winterfell and Starfall, robbing Edric of his birthright and causing a crisis between Riverrun and Winterfell. The alliance between the Riverlands, the Vale, and the North were secured by Hoster Tully's marriage of his daughters to those families. The entire alliance would be severely compromised if Hoster Tully found out that he had married his daughter to a man whose older brother had a living, legitimate son, as their children would not be the Lords of Winterfell but an auxiliary line.So, what happened was this: Ned and company kill the Kingsguard at the Tower of Joy, take Dany from a dying Lyanna while promising to protect her, and head to Starfall on the pretense of returning Dawn to Ashara Dayne. Once there, he does an exchange: Daenerys for Jon. The Daynes, with their strong connections to what remains of the Targaryen political apparatus, smuggle Danaerys out of the country and to Viserys, who lies to her about her past in order to shore up his own claims to the throne and then parades her about as his sister. Dany maintains puzzling memories of growing up around lemon trees and wooden beams (Dorne). Edric's claim to Starfall is secured, and he is fed the lie that Jon Snow was his milkbrother in order to erase suspicion. Ashara, devasted by the loss of her brother, her lover at the hands of Aerys, and now her son (the last thing she has to remember Brandon by) kills herself. Without this context, the death of Ashara simply does not make sense. Neither does the high regard which the house holds Ned in, who reputedly dishonored the famously beautiful Ashara and then drove her to suicide by killing the house's most prodigal son.