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.
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.
The main antagonist could also be an ancient and powerful spirit from the underworld that was sealed away long ago. Percy's presence as the son of Hades accidentally awakens this spirit. The spirit then starts to spread darkness and despair throughout the world of the demigods, and Percy has to find a way to stop it before it's too late.
Another option could be some rogue demigods who are jealous of the son of Hades' powers. They might form a group and try to cause trouble for him and Percy Jackson. They could be working with some minor gods who have a grudge against Hades, creating a complex web of enemies.
One possible plot could be Percy struggling with the duality of his parentage. He might face discrimination from both the children of Hades and Poseidon. For example, the children of Poseidon could see him as a threat because of his connection to Hades, and vice versa. He could then go on a journey to prove that his dual heritage doesn't define him, but rather gives him unique powers to protect the world of the gods and mortals.
Common themes include acceptance. The OC has to find a way to be accepted by the other demigods who may be wary of him because of his father. Redemption is another theme sometimes. Maybe the OC has made mistakes or has been associated with 'evil' because of Hades, and they strive to prove themselves. And the theme of fate also pops up often. They wonder if their destiny is set because of being the son of Hades or if they can change it.
There are definitely some. However, the quantity might be less compared to more common pairings in the Percy Jackson fanfiction world. One great aspect of these fanfics is how they often blend the themes associated with Hades and Hestia. For instance, Percy might struggle with the duality of his nature - the brooding, dark side from Hades and the nurturing, home - like quality from Hestia. It makes for a very interesting read if you can find well - written ones.