One idea could be Percy having to mediate between Zeus and Hades during a major feud. Since he's the son of both in this fanfiction, he has a unique connection to them. He could use his powers and his knowledge of both of their domains to try and broker peace.
In a fanfiction, Percy as the son of Zeus and Hades could have a very conflicted personality. He would be torn between the light and airy nature of Zeus's domain and the dark and brooding nature of Hades's. This internal struggle could be a major part of his character arc, as he tries to find a balance between the two aspects of his heritage.
One great fanfiction idea could be a story where Zeus, Poseidon and Hades have to team up to face a common threat that is even more powerful than them. Maybe a primordial being awakens and they must put aside their differences. In this story, Percy could be the mediator between them, using his unique position as the son of Poseidon to try and keep the peace among the three powerful gods.
A fanfiction could be about a competition among Zeus, Poseidon and Hades, and Percy has to be the judge. The competition could be something like who can best help a small, struggling mortal town without using their full power. This would show a different side of the gods and also test Percy's ability to be fair and objective.
Yes, there are quite a few interesting ones. You can start by looking on fanfiction platforms like Archive of Our Own. Some fanfics might explore Percy's relationship with his divine parents in unique ways, or perhaps put him in new and exciting adventures that are different from the original series.
Annabeth Chase might also be an ally. Her intelligence and strategic thinking could be valuable assets for the son of Hades. She could help him plan his moves, especially when dealing with the complex politics among the demigods and gods.
One possible plot could be Percy's struggle with his new identity as the adopted son of Zeus. He might face envy from other demigods who don't understand why he was chosen. He could also have to deal with Zeus' overbearing nature as he tries to train Percy in godly powers. And there would likely be new quests that involve both his ties to Poseidon and his new relation to Zeus.
If Naruto was the son of Zeus in a Percy Jackson fanfiction, his character would have to adapt to a whole new world. The demigod world of Percy Jackson has its own rules and politics. Naruto, being used to the Hidden Leaf Village's system, would have to learn about cabins, quests, and the whole Greek mythology aspect. His shadow clones could be used in really creative ways during quests, like scouting ahead or fighting multiple monsters at once. And his ability to never give up would be a great asset when facing the many challenges that come with being a demigod.
One possible plot could be Percy discovering his new parentage of being the son of Zeus and Hera. He might struggle with the new identity as it goes against what he previously knew. For example, he could face challenges from the other demigods who are suspicious or jealous of his new status. He then has to prove himself worthy of this new heritage while also dealing with the complex politics among the gods.
It could be a powerful Titan who escaped from Tartarus and wants revenge on Zeus by targeting his son Percy. This Titan might have unique powers that make him a formidable foe.