Zeus is also often paired with Percy. Their relationship is always complex and full of tension in the original stories, so fanfiction can explore this further. Maybe Zeus asks Percy to do a special task for him in order to earn the gods' full acceptance of Percy.
Another idea is a love story where Percy falls in love with a minor god or goddess. This could involve them having to keep their relationship a secret from the other gods because it's forbidden. The story could explore the challenges they face in their relationship and how they deal with the jealousy and interference from other deities.
You can try fanfiction websites like Archive of Our Own (AO3). It has a large collection of fanfictions, and you might be able to find some great 'percy x gods' ones there. Just use the search function on the site.
Probably Poseidon, as he is Percy's father and has a vested interest in his son's well - being. Zeus might also be protective in some cases, despite their past differences, as Percy is a hero who often saves the world which benefits Zeus's realm as well.
Well, without reading the specific fanfiction, it's hard to say exactly who Percy is. But generally, in a 'champion of the gods' scenario, Percy would be a central figure. He might be a hero with unique powers, like being able to control certain elements or having superhuman strength. He could be a character who has to face various challenges and tasks on behalf of the gods, kind of like a chosen one in a fantasy story.
Zeus is a possibility. As the king of the gods, he might see Percy's potential and adopt him to have an ally among the demigods. Another could be Poseidon, since Percy is already his son in the original story. It would be an interesting twist to have Poseidon formally adopt Percy in a more divine way.
The gods might show their protection in various ways. For example, they could directly intervene in battles Percy is in, using their divine powers to shield him from harm. They might also send him signs or omens to guide him away from danger.
Another plot might involve Percy being accidentally transported to a time when the male gods were still young. He has to figure out how to get back to his own time while also getting involved in the gods' early adventures. He could end up changing some small things in the past that have unexpected consequences in his present time.
Zeus is often one of the main gods showing possessiveness. He sees Percy as a powerful asset or a potential threat, so he wants to keep Percy in check. Poseidon, as Percy's father, also shows possessiveness in a different way. He wants to protect Percy but also use his powers for his own purposes sometimes.
One idea could be a fanfiction where Percy uses his powers in a new and unique way to solve a problem that the gods themselves are struggling with. For example, he might find a way to stop a magical plague that is affecting both mortals and immortals. This would show his value and ingenuity, which the gods might appreciate.