One possibility could be that Percy, being the banished son of Odin, has to prove himself in the world of Percy Jackson. Maybe he has powers from Odin that are not fully understood in the demigod world. He could go on a journey to discover his true identity and find a way to regain his place. For example, he might face challenges from both the Norse and Greek mythological beings. He may have to use his wits and newfound powers to overcome these obstacles.
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 in the fanfic, there's a prophecy that gets misinterpreted. The gods think Percy is going to cause great harm based on this wrong interpretation, so they send him to Tartarus. It could also be that there's some sort of power struggle among the gods. Percy might be caught in the middle of it, and one side decides to get rid of him by banishing him to Tartarus to tip the balance in their favor.
First of all, Percy's resourcefulness would come into play. He would probably try to make allies among the less - hostile creatures in Tartarus. He has a way of befriending those who are often overlooked. Also, he would use his knowledge of Greek mythology. Since he knows about many of the monsters and their weaknesses from his studies, he can use this knowledge to survive. His connection to his friends, even if they are far away, might also give him the strength and motivation to keep going and find a way to escape or at least endure.
One popular 'Percy Jackson Banished' fanfiction might be 'The Banished Hero's Return'. It often explores Percy's journey after being banished, facing new challenges and perhaps reuniting with old friends in unexpected ways. Another could be 'Banished to the Shadows' which delves into the darker aspects of Percy's banishment and how he copes with being an outcast in the world he once knew.
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.
One popular 'Percy Jackson Banished' fanfic might be 'The Banished Hero's Return'. It often has Percy facing new challenges outside of Camp Half - Blood and having to prove himself all over again. He might encounter new mythical creatures and make unexpected allies.
His powers would likely be a blend of his original demigod abilities and those of Odin. He may still retain some connection to water, but it could be transformed. Maybe he can turn water into ice like the Norse frost giants. Also, he could have Odin's gift of knowledge, which means he could understand ancient languages or know things about the past and future without being taught. This would give him an edge in battles and quests compared to the original Percy.
Conflict is a very common theme as well. There are often conflicts between Percy and the new characters he meets, either because of misunderstandings or because they have different goals. And there's also the overarching conflict of Percy trying to find a way back home or to reverse his banishment. This conflict drives the plot forward and keeps the story exciting.
Another option could be Wattpad. Although it has a wide range of content, including many fanfictions. You can search for the specific 'Percy Jackson Banished' fanfiction there. Just be aware that since it has a lot of different types of stories, it might take a bit more effort to find exactly what you're looking for compared to more specialized fanfiction platforms.