Maybe it's Zeus. If Zeus becomes too tyrannical and starts to oppress the other gods and demigods, Percy could find a way to banish him. This would be a very big deal as Zeus is the king of the gods. It would lead to a power struggle among the other gods to fill the void.
It could be Ares. He is often portrayed as a violent and unpredictable god. Percy might banish him if he goes too far in causing wars and chaos among mortals. His banishment would have a huge impact on the battlefields of the mortal world.
Well, perhaps the god in question has been misusing their powers and causing harm to mortals and demigods alike. Percy, being the hero he is, decides to take matters into his own hands and banish the god. After the banishment, Percy then has to find a way to fill the void left by the god. He might have to take on some of the god's former responsibilities, which is no easy feat as he is still just a demigod.
Obviously, Percy is one of the main characters since the title implies it. He would be the central figure whose story the fanfiction revolves around.
One possible interesting plot could be Percy's struggle to adapt to his new status as a banished god. He might have to deal with the loss of his former powers and the isolation from his old world.
A possible betrayer could be a jealous demigod. There are always those in the Percy Jackson world who envy Percy's abilities and the attention he gets. This demigod might see an opportunity to get rid of Percy and make themselves look better in the eyes of the other campers.
It could be a rogue demigod who has a vendetta against Percy. This demigod might have spread the false rumors that led to Percy's banishment in the first place, and now he wants to make sure Percy never returns. He uses his own powers and manipulates other demigods and monsters to achieve his goal.
One possible plot could be that in a world where the gods of Percy Jackson and the Avengers co - exist, Percy is somehow tricked or forced to banish the Avengers. Maybe there's a power struggle among the gods and they see the Avengers as a threat to their power, so they use Percy, who has a strong connection to the divine world, to get rid of them. Percy might be conflicted as he may admire the Avengers' heroism but has to follow the gods' orders.
In some fanfictions, Percy might be banished because he goes against the traditional ways of the demigod society too much. For example, if he tries to form an alliance with a group that the other demigods consider enemies. His actions could be seen as a threat to the stability of their world, so they banish him to keep the peace.
Well, he might face a lot of challenges. Alaska is a cold and harsh place. He could encounter new mythical creatures that are native to that region in the fanfiction. Maybe he has to find a way to survive the extreme cold while also dealing with any enemies that banished him there in the first place.
Another possibility is a group of outlaws or rebels in the banished world who don't trust Percy because he's from another place. They might think he's been sent to spy on them or take over. So they would constantly try to get rid of him. They could be very resourceful and know the ins and outs of the new world better than Percy, which would make them tough opponents.
Obviously, Percy Jackson is the central character. He is the one who is banished and has to deal with the consequences. His friends like Annabeth Chase might also play important roles. She could be trying to find a way to bring Percy back or dealing with the fallout of his banishment at Camp Half - Blood.