He'd probably be more brooding. Growing up with Hades, he'd be exposed to the darker side of the mythological world constantly. So, instead of the somewhat upbeat Percy we know, he might be more withdrawn and moody.
One possible plot twist could be that Percy has a different set of powers. Since Hades is the god of the underworld, Percy might have an affinity for shadow manipulation or communicating with the dead in a more in - depth way than in the original stories. For example, he could be able to summon undead warriors to fight for him.
In fanfiction, Percy Jackson as the champion of Hades could bring a whole new dynamic. He might have to deal with the darker aspects of the Underworld. For example, he could be in charge of ensuring the souls are properly sorted, which is a huge responsibility.
Well, it would be a very dark and different story. Percy might grow up with a different set of values. He could be more in tune with the Underworld's rules and power dynamics. Maybe he'd have a more cynical view of the world above. And his relationship with the other gods would be strained from the start as he is associated with Hades.
The story might focus on her struggle to fit in with the other demigods. Since she was raised by Hades, they might be suspicious or even afraid of her. But she could prove herself to be a hero in her own right. She could also have a unique relationship with the other children of the Big Three (Zeus and Poseidon).
One possible plot twist could be that she discovers she has a different parentage than she thought. Maybe Hades isn't really her father, but someone else who had a reason to leave her with Hades. Another twist could be that she has to fight against Hades at some point, even though he raised her.
One common theme is the exploration of Hades' character. Many fanfictions try to humanize him, showing that he's not just the 'bad guy' as he's often portrayed in the original stories. For example, they might show his love for his family in a different way.
One common theme is the exploration of power dynamics. Since Hades is a powerful god and Percy has his own set of powers, fanfictions often play with how they interact and who has the upper hand in different situations. Another theme is family relations. Hades has a complex family, and Percy's own family situation is also quite intricate. Fanfictions might explore how their family backgrounds affect their relationship and actions.
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 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.