He dreamt an old dream, of three knights in white cloaks, and a tower long fallen, and Lyanna in her bed of blood. - AGoT, Eddard X (ToJ)
She dreamt an old dream, of three girls in brown cloaks, a wattled crone, and a tent that smelled of death. - AFfC, Cersei VIII
In the dream his friends rode with him, as they had in life. Proud Martyn Cassel, Jory’s father; faithful Theo Wull; Ethan Glover, who had been Brandon’s squire; Ser Mark Ryswell, soft of speech and gentle of heart; the crannogman, Howland Reed; Lord Dustin on his great red stallion. Ned had known their faces as well as he knew his own once, but the years leech at a man’s memories, even those he has vowed never to forget. [...]
They were seven, facing three. In the dream as it had been in life. - Ned
They were three in the dream, as they had been in life. Fat Jeyne Farman hung back as she always did. It was a wonder she had come this far. Melara Hetherspoon was bolder, older, and prettier, in a freckly sort of way. - Cersei
In the dream they were only shadows, grey wraiths on horses made of mist. - Ned
In the dream the pavilions were shadowed, and the knights and serving men they passed were made of mist. - Cersei
A storm of rose petals blew across a blood-streaked sky, as blue as the eyes of death. - Ned
But in the dream her face dissolved, melting away into ribbons of grey mist until all that remained were two squinting yellow eyes, the eyes of death. - Cersei
Two more pieces that stood out to me from that Cersei chapter, though they don't seem to have any direct counterparts to Ned's dream:
In the dim green tent, the blood seemed more black than red. Maggy’s toothless mouth trembled at the sight of it. “Here,” she whispered, “give it here.” When Cersei offered her hand, she sucked away the blood with gums as soft as a newborn babe’s. The queen could still remember how queer and cold her mouth had been.
You stupid girl, the queen thought, angry even now. Jaime does not even know you are alive. Back then her brother lived only for swords and dogs and horses... and for her, his twin.
The newborn babe reference from the first paragraph is obvious enough, but swords, dogs and horses – does anybody remember where that parallel can be found? It's a Last Hero reference.
“He set out into the dead lands with a sword, a horse, a dog, and a dozen companions.”
“One by one his friends died, and his horse, and finally even his dog, and his sword froze so hard the blade snapped when he tried to use it.”
- AGoT, Bran IV (Old Nan's tale of the Last Hero)
The linguistic similarities between these two dreams are undeniable. But why? What is GRRM trying to tell us? It took me a while to figure it out, but I think I did.
The trick was to view Cersei's dream as the template. This is a counter intuitive idea, since we read about Ned's ToJ dream in the first book. But once you do this the next step is to ask what was going on in her dream?
Cersei was dreaming of a prophecy told to her by Maggy the Frog, who is described as a woods witch in Cersei I. This was my Aha! moment, because the series contains at least one other known woods witch prophecy.
“Your [Dany's] grandsire commanded it. A woods witch had told him that the prince was promised would be born of their line.”
- Ser Barristan from: ADwD, Daenerys IV
So, my conclusion is that Ned's ToJ dream is more than we thought it was. Not only does it convey to us information about Jon's parentage and legitimacy – though let's not argue about that here – it's a dream about the time in which Ned's life became intertwined with the prophecy of the Prince that was Promised.
Now, many have already connected Promise me, Ned to tPtwP prophecy, so that's nothing new. The addition here, is that the events at the ToJ are tangled up in prophecy, and that's what GRRM was trying to tell us with the linguistic similarities he used in both dreams.
I'll leave you with the final lines of Ned's ToJ dream:
“I promise,” he whispered. “Lya, I promise…"
Special thanks to butterbumps!, Lady Gywn, and Ygrain for their assistance.