Well, it could be about Percy learning to survive in the wild using his demigod abilities in unique ways. Maybe he discovers new mythical creatures that are only found in the wild areas he's in.
Obviously, Percy Jackson would be the main character. But there could also be some wild spirits or minor gods that take on a mentoring role for him, like a dryad who teaches him about the plants in the forest.
Maybe Percy would grow up with the instincts of a tiger. He could be super fast and stealthy like them. And perhaps he'd have a special bond with the tigers that would help him in battles against monsters, just like how his demigod powers usually do.
Obviously, Percy Jackson himself would be the main character. The tigers that raise him would also be important characters. They could be like his family in the story.
Poseidon is a likely candidate. Since he is Percy's father in the original series, it would make sense for him to take a more active role in raising Percy. He could teach Percy all about the ocean, its secrets, and how to use his water - related powers more effectively.
Well, it could be that Percy is raised by a specific god like Poseidon but in a more direct and involved way. Maybe the god teaches him all kinds of super - secret powers and knowledge that normal demigods don't get. For example, he might be taught how to control water in ways that are far beyond what he learned at Camp Half - Blood. This could lead to him having a whole new set of adventures, facing different kinds of monsters and challenges that are related to the god's domain.
Perhaps in this fanfic, Percy is raised as a sort of prince on Olympus. He could be given tasks by the gods to prove his worthiness to be among them. This might include going on quests that are not just for the safety of the mortal world, but also for the stability of Olympus itself. He would also have to navigate the complex social hierarchies among the various gods and goddesses.
A possible new character could be an oracle - like figure who is not from the traditional Delphi but is a seer on Olympus. This character could give Percy cryptic prophecies that are related to the future of Olympus and the demigod world. There might also be a group of centaurs who are specifically trained to protect Percy and serve as his advisors, different from the ones at Camp Half - Blood.
He would likely be more cautious and distrustful. Since he's being raised by a monster, he may not have the same sense of heroism and justice that the normal Percy has. He might see the world as a more dangerous place where only the strong survive.
His character would become more confident in his artistic abilities. Apollo is the god of many arts, so Percy would likely be exposed to music, poetry, and painting. He might be more flamboyant in his actions and speech, as Apollo has a certain charm. For example, he could start using more flowery language when talking to his friends or enemies.