One interesting plotline could be Percy adjusting to the Underworld. He might have to learn new rules and customs, like how to interact with the various spirits and creatures there. Another could be Hades trying to train Percy in unique ways, perhaps teaching him powers related to the Underworld that he didn't have before.
Obviously, Percy and Hades will interact a great deal. Percy will also likely interact with the Furies, as they are servants of Hades. They could be like his protectors or trainers in some sense. Another character could be Nico di Angelo, since he has a connection to Hades as well. They could bond over their shared relation to Hades.
In non - lemon fanfiction, Percy is often depicted as brave and resourceful as in the original series. His loyalty to his friends and his struggle with his identity as a demigod are common themes. Hades, on the other hand, is usually portrayed as a complex figure. He is powerful and mysterious, with his own motives and a sense of duty towards the underworld. Their relationship might be explored in terms of mutual respect or a begrudging alliance.
One idea could be that Percy discovers a demigod child who has been abandoned in the mortal world. He decides to adopt the child to protect them from monsters and to give them a proper home at Camp Half - Blood. He then has to teach the child about their demigod heritage and how to use their powers.
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.
Percy could also become angry. He values his freedom and his relationships with his friends and family. Hades' possessiveness is a threat to all of that. So he might stand up to Hades, even if it's a risky move. He'll tell Hades that he won't be controlled or owned. He'll fight back using his powers and his wits. He might also seek help from other gods or his demigod friends who have had experiences with Hades.
You can start by giving her a distinct personality. Maybe she's shy because she's always been feared for her father's sake. Then, gradually show her growing confidence as she discovers her own strengths. For example, if she has the power to raise small skeletons as minions, at first she's scared of it, but later she learns to use it for good. Also, explore her relationships with other characters. Her friendship with Percy could be based on mutual understanding of being different among the demigods.
One common theme is mentorship. Apollo often takes on the role of a mentor to Percy, teaching him about different aspects like new powers or knowledge. For example, in many stories, Apollo teaches Percy about the arts which Percy is not very familiar with initially.
Maybe Percy discovers a kid in a difficult situation, like being an orphan in a monster - infested area. He decides to take the kid under his wing to protect him from harm.