What was Dany's vision in the House of the Undying?
Back in Season 2, when Daenerys' three dragons were tiny little balls of CGI cuteness and not giant, goat-eating, civilian-burning nightmares, they figured in a scheme that saw Xaro Xhoan Daxos team up with the warlock Pyat Pree to stage a coup that installed Xaro as king of Quarth. Along the way, the two stole Dany's dragons and stashed them in the House of the Undying, home of the Quarth warlocks and a generally spooky, windowless place.
Naturally, Dany decides to get her kids back, despite Ser Jorah's warnings that she'll probably have to take on a good deal of magic and sorcery if she hopes to succeed. "What about my magic?" Dany responds, and Jorah has no choice but to say, "Sure, right, you're magical too, you've got so much magic," before they head off into the den of warlocks.
Dany begins to have a series of visions as she follows the screeching sound of her dragons. First, she walks into the Red Keep, except the roof has been destroyed, possibly burned, and it's snowing. She approaches the Iron Throne and nearly touches it before the noise calls her into another vision north of the Wall, in which she reunites with her husband, Khal Drogo, and their stillborn child.
Some online communities have tried to argue that the substance falling in the throne room isn't snow, but ash -- unfortunately, this doesn't fit with what appears on screen (you can see water dripping off Dany's hand at one point), and both Emilia Clarke and and director Alan Taylor call the stuff snow. Plus, given what we know about how the show unfolded, it's unlikely that David Benioff and D.B. Weiss envisioned this particular scene as a direct reference to the series finale.
The visuals of the finale clearly mirror the visuals of the House of the Undying, with white, snowy debris falling during every scene. When Daenerys approaches the Iron Throne, white ash falls around her, and this time she actually touches the seat. She clearly believes that this is the fulfillment of her destiny.
But hang on a second! What if "snow" isn't literal snow? Isn't there someone... important... named... Snow? There is: It's Jon Snow, duh. Snow in the throne room of the Red Keep, which has been destroyed? Dany almost seizes the throne in her vision, but stops just short? Somebody get out the yarn board, because it's all connected.