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.
Perhaps the fanfic is about Percy's discovery of his true heritage as the son of Odin. He might be banished because of some prophecy or a power struggle within the Norse pantheon. In the Percy Jackson world, he has to hide this identity at first. But as the story progresses, he realizes he can't keep it a secret. His journey could involve making allies among the demigods who accept him for who he is, and together they face a common enemy that is somehow related to his banishment.
The plot could revolve around Percy's struggle with his banishment. Since he is the son of Odin in this fanfic, he might be shunned by the other demigods at first. But as he shows his unique abilities, like Odin's wisdom or his ability to see things others can't, he gradually earns their respect. He could be on a quest to find a way to break the banishment curse. This could lead him to different realms, both in the Percy Jackson universe and those related to Norse mythology.