One possible plot could be Percy being framed for a crime he didn't commit by the Olympians. Maybe they needed a scapegoat for something that went wrong in the mortal world that they didn't want to take the blame for. Another plot might involve Percy discovering a dark secret about the Olympians that they don't want him to reveal, so they turn on him to keep him silent. It could also be that a new powerful enemy has influenced the Olympians against Percy, perhaps by spreading false rumors or using some form of mind control on them.
Obviously, Percy Jackson is the central character. The Olympians such as Zeus, Poseidon, and Athena are likely to be involved as the ones doing the betraying. Other demigods like Annabeth Chase could also be part of the story. She might be torn between her loyalty to Percy and her respect for the Olympians. There could also be minor gods and goddesses who either side with Percy or the Olympians, depending on their own motives and relationships. For example, Hades might see an opportunity to gain more power if Percy is out of the picture, or he could help Percy out of a sense of fairness.
Ares could be a likely character. He's always been somewhat of a wild card in the Percy Jackson series. He might betray Percy for his own amusement or to gain more influence in Olympus. Another could be Zeus. He's often paranoid about his power and might see Percy as a threat and betray him to protect his throne.
One common plotline is Percy finding out that the Olympians have been using him all along. For example, they might have sent him on dangerous quests not just to save the world but also to gain more power for themselves. Another could be Percy being betrayed by a close friend or ally who is working for the Olympians' self - serving plans. Maybe they set him up to take the fall for something he didn't do.
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.