Perhaps an ancient and dark deity who has been awakened due to some event related to Percy's exile. This deity wants to bring chaos to the world of gods and demigods, and Percy, even in exile, is the only one who can stop it. The deity would use all kinds of dark magic and minions to oppose Percy and his friends.
Maybe a rogue demigod who blames Percy for something and wants to make his exile permanent. This demigod could have some dark powers or be in league with some evil forces. They would constantly try to thwart Percy's attempts to return or find a way to clear his name.
One idea could be to explore Percy's life in exile and how he copes with being away from his friends and the world he knew. Maybe he discovers new powers or forms unexpected alliances. Another could be that he's exiled to a realm filled with ancient and forgotten creatures, and he has to find a way to survive and potentially return home.
One interesting plotline could be Percy trying to find a way to return home while facing new and unexpected challenges in his exile. He might encounter strange creatures or make new allies.
There could be a local guide in the place where Percy is exiled. This character could be an outcast from their own community and help Percy navigate the strange new land.
I'm not aware of any such fanfiction. The Percy Jackson fandom is generally centered around positive themes like the hero's journey, the power of the demigods, and their relationships. A story about Percy hanging himself goes against the very essence of what the character represents. He is a strong, resourceful hero, and most fanfiction writers would rather explore his adventures, his relationships with Annabeth and his friends, or his battles against the Titans and other foes.
One idea could be Percy being exiled to a forgotten part of the mythological world, like a lost island where ancient and dangerous creatures lurk. He has to learn their ways to survive and find a way back to his friends.
It could also be due to a sense of guilt. If something bad happened that he blames himself for, like someone getting hurt because of him, he might exile himself as a form of self - punishment. He'd rather be alone and deal with his own pain than face those he feels he's let down.
Maybe he feels guilty. Suppose he made a big mistake, like accidentally leading an enemy to a safe haven of demigods. The guilt could be so overwhelming that he exiles himself to punish himself and keep others safe from his potential future blunders.
It would be very shocking. Percy is a beloved character, and such a fanfiction would be a huge departure from the normal, positive portrayals. It could have a negative impact on younger fans who look up to Percy as a hero.