Maybe he feels betrayed by the Olympians. They often have their own agendas and don't always consider the well - being of demigods like Percy. For example, they might send him on dangerous quests without proper support.
His friends might be divided. Some who are more inclined to support Olympus might be at odds with Percy. For example, Annabeth, who has a certain respect for the gods, could face a dilemma in choosing between Percy and Olympus.
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.