Fic idea that I may or may not ever write #1: Zuko in Westeros
Feel free to use any or all of these ideas in your own fic if you want.
Shortly after the events of Zuko Alone, Zuko sees a flash of blinding white light and falls unconscious. In the Riverlands, around the time Robb is returning from his campaign in the West, Beric, Thoros, and the Brotherhood Without Banners are marching along a back road when there's a flash of of blinding white light. Once they blink the spots out of their eyes, they see a ring of fire and lying unconscious inside it is a certain scar-faced teenager. Seeing as he appeared accompanied by fire and light, they immediately assume he was sent by the Lord of Light and is special somehow.
Their certainty only increases when Zuko wakes up and reveals 1) that he's a firebender 2) he's the son of the Fire Lord and Fire Lady and a prince, and 3) he knows the Lord of Light/R'hllor by a secret name, which they take to be a super-special secret name as of yet unrevealed to the unworthy masses (Zuko asks them if they are referring to Agni when they first mention the Lord of Light). Take all this together, they immediately dub him the Prince of Light and Champion of Agni/R'hllor/the Lord of Light, and swear their undying allegiance to him.
He tries convincing them that, yes, he's a prince, and yes, he can make fire with his hands, but he's not that special, but Beric and co just respond with some vague, significant sounding proverb about how the Lord of Light works in mysterious ways or something like that (Zuko yells something about them sounding too much like Uncle).
So Zuko is stuck with this band of outlaws following him proclaiming him a prince (some even think he must be Azor Ahai). Having figured out by now that he's definitely not in Kansas anymore, Zuko doesn't really know what to do, but taking care of the Smallfolk seems like a decent thing to be doing, right? Oh hey, who's this girl we've captured that reminds me of my little sister in some (scary) ways? Apparently someone important named Arya Stark. She's a king's little sister or something. And she really wants this Hound guy dead because he killed her friend, which, fair (also reminiscent of Azula). The Hound guy is being given a trial by combat to prove his innocence. Hmm. I bet I could take him.
Zuko fights the Hound instead of Beric. Between being a prodigy with his dual dao and also having firebending, he wins and kills the Hound for Arya. He throws up afterwards, this being the first man he's actually killed himself, and he and Arya have a discussion about their first kills (this girl has killed how many people? Definitely very Azula like). Zuko basically imprints on Arya and decides to help her out. So instead of just ransoming her to her brother, he asks her what she actually wants. After some discussion, she decides she does want to go see her brother and family again, so they set off for Riverrun. Except along the way they find out Robb's actually going to the Twins for a (Red) wedding, so they head there instead, arriving just in time to fish all the bodies of the river, including Arya's mother.
Zuko takes a look at the Twins and thinks, I managed to sneak into Pohuai stronghold, this place can't be much harder. Having conveniently had both his swords and his Blue Spirit mask on him when he got transported to Westeros, he puts them on and Blue Spirits all the prisoners the Freys took out of the Twins (and also sets the place on fire and kills a great many Freys in the process). (Arya wanted to come to, but Zuko put his foot down. This girl might be just as scary as Azula, but she's also even younger than Azula. No way. No how.)
And that's about as far as I've thought this through. Arya gets declared the new Queen of the North by her brother's rescued bannermen, Zuko decides to stick around to take care of her, and by extension the Brotherhood also now sign on with the Starks, although they continue to be not-quite in the chain of command and still go about lynching looters regardless of what house they're sworn to.