I don't know if I'll ever write this, but I had an idea for a fic where Yue was the Avatar. I think I already posted an outline for this, but I've put a little more thought into it.
Katara and Sokka head to the NWT looking to gather support for the SWT's war efforts (and to find someone to train Katara). I think they go directly there. They find the NWT closed off like in the show, but they get taken in because they are Chief Hakoda's children. It's essentially what happened in the show, except Yue is a waterbender, and Katara is sent off with her to have her learn to be a proper water tribe woman. They go off to the healing huts, and then Katara's face off with Pakku happens, but Yue pleads Katara's case saying that eventually she'll have to go home and wouldn't it be better if she could defend herself or something like that. Katara's connection with Kanna is discovered and Pakku agrees to train Katara (there will probably be more of an explanation of why Pakku's relationship with Kanna matters. I think Kanna was pregnant with Kya when she left and Pakku's world is rocked learning what happened to the daughter he never met).
The time line is more stretched out here, and Katara and Sokka are there for a few months. Katara learns more about her grandmother's time there from Pakku and Yugoda. Sokka and Yue have a romance, but she's still betrothed, so it's secret and angsty and all that good stuff. Six months in, Katara is a master, and Sokka and Yue are trying to figure out what to do. A pirate ship turns up, and Captain Lee and his first mate Mushi come to warn the NWT that the Fire Nation is preparing to attack. No one believes them until the ash starts falling. The fire nation attacks, and it's discovered that Yue is the Avatar. The FN soldiers are beaten- mostly killed, but a few get away. Katara is furious with Pakku and Arnook, and the NWT in general for hiding this. She makes a big speech about how the world needs the Avatar and eveyone is fighting and dying and trying to stop the FN from destroying the world, but they were keeping the one person who could change the tide hidden out of selfishness. She and Sokka are, of course, kicked out. Lee and Mushi offer them passage on their ship. Sokka goes to say goodbye to Yue and she asks him to stay, and that she thought she could convince her father to let her marry him instead. He tells her he needs to go help his people fight. They kiss and Sokka and Katara prepare to leaved. The next day, as they're waiting for Captain Lee to send a row boat for them, Yue appears, bag in hand and tells them that she was tired of hiding and she wants to do her duty as Avatar, starting with mastering more waterbending than healing. She tells them that her father agreed, and they all leave with Captain Lee.
There's friction with Katara and Lee immediately. She thinks he's hiding something. He thinks she's a busybody who doesn't know when to stop prying. But, they work well together, and a friendship develops over their travels. Katara helps Yue master waterbending, and then they set off in search of an earthbending master. That's when they find out that Yue lied. Her father is furious and sending people to bring her home, and to bring Katara and Sokka back to face kidnapping charges. The Fire Nation now also knows that Yue is the Avatar, and send Azula to find her. Everyone converges on the group at once, and that's how it's discovered that Captain Lee is the disgraced Prince Zuko.
That's it. That's as far as I got. I don't know if this fic will ever see the light of day, but if I were going to do a non-Aang Avatar, this would be it.