One possible plot could be that Percy jumps off Olympus in a moment of extreme despair. Maybe he feels betrayed by the gods and their constant manipulation. He jumps, thinking it's an escape, but then discovers he has new powers or a new mission on earth. For example, he might land in a mortal city and realize he has to protect it from a new threat that the gods are ignoring.
Annabeth could be a main character. She might be desperately trying to find Percy after he jumps off Olympus. She could use her wisdom and strategic skills to track him down, facing various obstacles along the way, like mythical beasts or other unfriendly demigods. Another character could be Grover. He could sense Percy's distress and join Annabeth in the search, using his nature - related powers to help them.
Grover might also be an ally. His connection to nature could be useful in Percy's new role. For example, he could help Percy understand any nature - related aspects of Olympus or deal with nature spirits in the realm. Also, some of the more benevolent gods like Poseidon (Percy's father) and Athena (Annabeth's mother) might side with Percy, at least to some extent, as they have their own interests in seeing Percy succeed in maintaining order in Olympus.
Well, in this 'percy is raised on olympus fanfiction', it might focus on Percy having to adapt to the unique environment of Olympus. He would be surrounded by powerful deities. It could show his training, his education in the ways of the gods, and his adventures within the realm of Olympus, all while being raised there.
Maybe he has some bad experiences with the Olympian gods in the original story, and in fanfiction, those negative feelings are amplified. For example, he might be tired of the gods' constant interference and manipulation.
It's not likely that the actual Olympus (in the context of the Percy Jackson series being a place of the gods) reads fanfiction. Fanfiction is a creation of the fans in our world, and there's no indication in the books that the gods or Olympus as an entity would engage with it.
Perhaps the plot involves Percy using his new princely powers to bring about positive changes in Olympus. He could work on improving relations between the gods and demigods, for example. Since he has always been a bit of an outsider among the demigods because of his unique abilities, this new role could give him the opportunity to bridge the gap between the two groups. He may also have to face some sort of conspiracy within Olympus that threatens his position as prince.
A lesser - known but really interesting fanfiction is 'Annabeth's Quest for Wisdom'. This story follows Annabeth as she goes on a solo quest to gain more knowledge and wisdom. It shows a different side of her character, her determination and her thirst for knowledge in a very engaging way.
One great fanfiction is 'The New Quest of Percy Jackson'. It continues Percy's adventures in a very believable way, with new challenges and old friends by his side. The author really captures the essence of the original series.
Obviously, Percy would be one of the main characters. And probably the Olympian gods like Zeus, Poseidon, Athena etc. would also play important roles as he is raised in their domain.