This is a odd statement by Jaime. Catelyn is questioning him when he is captive at Riverrun and they are exchanging truths. He tells her what he witnessed of Brandon and Rickard's deaths.
After Gerold Hightower himself took me aside and said to me, "You swore a vow to guard the king, not to judge him." That was the White Bull, loyal to the end and a better man than me, all agree."
"Aerys ..." Catelyn could taste bile at the back of her throat. The story was so hideous she suspected it had to e true. "Aerys was mad the whole realm knew it, but if you would have me believe you slew him to avenge Brandon Stark ..."
"I made no such claim. The Starks were nothing to me. I will say, I think it passing odd that I am loved by one for a kindness I never did, and reviled by so many for my finest act.
"I've never lain with any woman but Cersei. In my own way, I have been truer than your Ned ever was. Poor old dead Ned. So who has shit for honor now, I ask you? What was the name of that bastard he fathered?
Catelyn took a step backward. "Brienne."
"No, that wasn't it." Jamie Lannister upended the flagon. A trickle ran down onto his face, bright as blood. "Snow that was the one. Such a white name ... like the pretty cloaks they give us in the Kingsguard when we swear our pretty oaths."
I found some interesting stuff that might say something about the Fisherman's Daughter story. I think Jon knows something on a subconscious level; that he may have gleaned some informaton from his dreams. I'm also not sure if it's Ned who is giving him the dreams. Recall that Bran and Rickon have the same dream of their father in the crypts and Jon's recurring dreams of going down to the crypts.
A Storm of Swords - Jon VI "Drink this." Grenn held a cup to his lips. Jon drank. His head was full of wolves and eagles, the sound of his brothers' laughter. The faces above him began to blur and fade. They can't be dead. Theon would never do that. And Winterfell . . . grey granite, oak and iron, crows wheeling around the towers, steam rising off the hot pools in the godswood, the stone kings sitting on their thrones . . . how could Winterfell be gone? When the dreams took him, he found himself back home once more, splashing in the hot pools beneath a huge white weirwood that had his father's face. Ygritte was with him, laughing at him, shedding her skins till she was naked as her name day, trying to kiss him, but he couldn't, not with his father watching. He was the blood of Winterfell, a man of the Night's Watch. I will not father a bastard, he told her. I will not. I will not. "You know nothing, Jon Snow," she whispered, her skin dissolving in the hot water, the flesh beneath sloughing off her bones until only skull and skeleton remained, and the pool bubbled thick and red. insert quote here
In one of Ned's dreams of Lyanna, she weeps tears of blood;
He was walking through the crypts beneath Winterfell, as he had walked a thousand times before. The Kings of Winter watched him pass with eyes of ice, and the direwolves at their feet turned their great stone heads and snarled. Last of all, he came to the tomb where his father slept, with Brandon and Lyanna beside him. "Promise me, Ned," Lyanna's statue whispered. She wore a garland of pale blue roses, and her eyes wept blood. Eddard XIII
The eyes weeping blood is mirrored in Jon's dream of Gilly as well as a baby swap;
A Dance with Dragons - Jon II – baby swap Whether the wrong-way rangers would return was another question. Knights they might be, but they did not know the north. There will be eyes along the kingsroad, not all of them friendly. It was none of Jon's concern, though. Let Stannis have his secrets. The gods know that I have mine. Ghost slept at the foot of the bed that night, and for once Jon did not dream he was a wolf. Even so, he slept fitfully, tossing for hours before sliding down into a nightmare. Gilly was in it, weeping, pleading with him to leave her babes alone, but he ripped the children from her arms and hacked their heads off, then swapped the heads around and told her to sew them back in place. When he woke, he found Edd Tollett looming over him in the darkness of his bedchamber. "M'lord? It is time. The hour of the wolf. You left orders to be woken."
A Dance with Dragons - Jon III Brothers, oh brothers, my days here are done, the Dornishman's taken my life, But what does it matter, for all men must die, and I've tasted the Dornishman's wife!
Val stood on the platform as still as if she had been carved of salt. She will not weep nor look away. Jon wondered what Ygritte would have done in her place. The women are the strong ones. He found himself thinking about Sam and Maester Aemon, about Gilly and the babe. She will curse me with her dying breath, but I saw no other way. Eastwatch reported savage storms upon the narrow sea. I meant to keep them safe. Did I feed them to the crabs instead? Last night he had dreamed of Sam drowning, of Ygritte dying with his arrow in her (it had not been his arrow, but in his dreams it always was), of Gilly weeping tears of blood. Jon Snow had seen enough. "Now," he said.
Jon has sent Sam (who will be a maester) away with Gilly (a nursemaid) after swapping out the King's baby to send them into hiding and safe keeping. Rumors of the storm at Eastwatch, the drowning of the maester, Jon singing the Dornishman's Daughter out of the blue while he watches Rattleshirt (glamored as Mance) burned to death (Brandon and Rickard?).
This all echos back to the story of the Fisherman's daughter and Bran's comment that he spoke with his father in his dream of the crypt and it was more disturbing than anything he had dreamed before. They talked about Jon. I think Jon as the 3EC talked with his father as well and like Bran some of his dreams are half remembered or forgotten when he wakes.
Are some of the details spilling out with Sam and Gilly's baby swap?
In another of Ned's dreams of the crown of blue roses; he has blood on his hands and Lyanna weeps blood, just like Gilly.
Ned Stark reached out his hand to grasp the flowery crown, but beneath the pale blue petals the thorns lay hidden. He felt them clawing at his skin, sharp and cruel, saw the slow trickle of blood run down his fingers, and woke, trembling, in the dark