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.
One common theme is the struggle with identity. Since Percy is usually the son of Poseidon in the original series, making him the Son of Hades in fanfiction often leads to him dealing with a new sense of self. Another theme is power and its consequences. Hades has a lot of dark and powerful abilities, so fanfictions might explore how Percy copes with having access to such powers.
One great 'Percy Son of Hades' fanfiction is 'The Underworld's Hero' on FanFiction.net. It has an interesting plot where Percy, as Hades' son, discovers new powers and faces challenges in a different light compared to the original series. The character development is really well - done.
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.
Obviously, Percy Jackson himself is a main character. He's the one with the new identity as the son of Hades. Nico di Angelo could also be a major character as he has a strong connection to Hades already. And Annabeth Chase would likely be involved too, as she's Percy's close friend and often by his side in adventures. There could also be some new characters created specifically for the fanfiction, like a powerful Underworld spirit that becomes Percy's guide or an enemy within the Hades - influenced areas.
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 common theme is the struggle with identity. Since Percy's parentage is so different from the original, he has to figure out who he really is. Another theme is power. With Hades and Artemis as his parents, he could have a whole new set of powers to discover and control. And there's also the theme of acceptance, both from himself and from the other characters in the Percy Jackson universe.
One idea could be Percy having to navigate the Underworld politics due to his relation to Hades. He might face challenges from other underworld creatures who don't trust him. Another could be Percy using his newfound powers as the son of Hades to fight against an ancient evil that threatens both the mortal and underworld realms. Or perhaps he has to deal with the stigma of being Hades' son in the demigod community and prove himself worthy.
Percy could have the power to manipulate the coldness that is often associated with the Underworld. He could freeze his opponents or create ice - like structures out of the cold energy. This would be different from his water - based powers in the normal story. Also, he may have a stronger connection to the gems and precious metals that are found in the Underworld, being able to sense them and even use them in some way for his benefit.
Another plot could involve Percy's relationship with his father, Hades. Maybe Hades has some grand plan for Percy that he doesn't fully understand at first. Hades could be trying to use Percy to gain more power in the world of the gods, but Percy resists and has to fight against his father's wishes while also dealing with the internal conflict of not wanting to completely turn his back on his family.