...so? They're uncle and nephew.
The point i'm making is that Eddard's dreams about his promise to Lyanna have nothing to do with Jon Snow, because Jon is not Lyanna's child.
If you read the earlier Ned chapters, he has in fact been thinking about him.
""For the first time in years, he found himself remembering Rhaegar Targaryen." - chapter 35, Eddard IX
Conversely, he doesn't think about Ashara Dayne at all, nor Brandon in any such capacity.
"they whispered of ser Arthur Dayne... Ned had carried Ser Authur's sword back to the beautiful young sister" <-- this is when Eddard picks up Jon on his way back to Winterfell.
"Catelyn had asked her husband the truth of it, asked him to his face."
"Never ask me about Jon" - Ned <-- Catelyn asks about Ashara being the mother and Ned simply states he never once her to question anything about Jon, so again: the question was phrased on Ashara, but Ned starts talking about Jon, because Jon is Ashara's baby.
"from that day on, the whispering had stopped, and Ashara Dayne's name was never heard in Winterfall again."
chapter 6, Catelyn II.
Dany being Lyanna's daughter makes absolutely no sense.
?
The rest of the castle garrison at Dragonstone would have known whether she was legit or not; Rhaella's pregnancy was not a secret.
And your source is Viserys, one of the biggest worms out there. How do we know Daenerys was born at Dragonstone? Remember, the author has purposely made Robert's Rebellion timeline into a mess.
Viserys has weakened Daenerys claim to the Throne and made her his own asset.
Daenerys has foggy memories about the House with the red door and a lemon tree in Bravoos, but there are no lemon trees in Bravoos. It's a port city. It's a cold and windy treeless place:
"I'm chilled to the bones. Where are the bloody orange trees? I always heard there we're orange trees in the Free Cities. Lemons and Limes." - Mercy, TWoW
And let us remember, that Quathe keeps rambling about Daenerys remembering who she is.
Asshai holds the truth!
And how and why would she even have come into the Targaryens' possession?
Well again, we know nothing about Daenerys past except she lived in a peaceful house with flowers, green grass, lemons and a red door.
It would certainly involve Ned Stark and whoever Quathe is.