Naruto, as the son of Hades, would have some really cool powers. His Rasengan could potentially be infused with the power of the Underworld, making it a much more deadly attack. He could also have the power of persuasion over the dead, which means he could make them tell him secrets or do his bidding. His ability to sense chakra in the Naruto world could be translated into sensing the life force or the 'essence' of beings in the Percy Jackson world. This would give him an edge in detecting enemies, whether they are living or undead.
Well, one way could be that Naruto, being the son of Hades, would have unique powers. He might be able to use his shadow clones in a new way, like sending them into the Underworld to gather information. His connection to Hades could also make him a target for other gods or demigods who are either afraid of Hades' power or want to use Naruto for their own gains. Maybe he would struggle to fit in at Camp Half - Blood at first because of his different origin compared to the normal demigods there.
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 idea could be Percy discovering new powers related to being the son of Hades. For example, he might be able to control the shadows in a unique way that no other demigod can. He could use this power to spy on enemies or to travel quickly through the underworld.
Maybe the son of Hades in the fanfic has a special bond with a particular type of monster that is usually associated with death, like a Cerberus - like creature. Percy Jackson and his friends are initially scared of this bond but then realize it could be used to their advantage. For example, the son of Hades could use this bond to control the monster and have it help them in a battle against a common enemy. The fanfic could also explore how the son of Hades tries to fit into the world of Camp Half - Blood while being different from the other demigods.
One way to develop the storyline could be to explore Percy's struggle with his new identity. Since Hades and Persephone are such complex deities, Percy might have a darker side to him. He could struggle with controlling powers related to the Underworld, like shadow manipulation or communicating with the dead. For example, in a school setting, he might accidentally let out some spooky manifestations when he's stressed.
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.
A major theme could be identity. Percy has to come to terms with being the son of two very powerful and different deities. It's not like being the son of Poseidon where he has a more straightforward heroic image. Here, he has to figure out who he really is.
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.