Well, one possible plot could be Percy and Nico having to deal with typical high school problems like bullies. Percy might use his water powers in a discreet way to stop them. Nico could be the mysterious loner at first but then make some real friends as he shows his cool powers. And they might also have to hide their demigod identities from the normal students.
In a fanfiction where they go to high school, Percy would likely be popular for his heroic deeds and good looks. He might join the swim team and break all the records. Thalia could be the cool girl who rides a motorcycle to school and gets into trouble for parking it in the wrong place. Nico would be the mysterious new kid. There could be a plot where they have to team up to stop a monster that is lurking in the school basement.
A powerful monster from the demigod world that has a vendetta against Percy and Nico could be the main antagonist. It disguises itself as a teacher or a student and slowly starts to manipulate things in the school. It tries to isolate Percy and Nico from their friends and then attack them when they are at their weakest.
Well, perhaps Percy and Nico could be transferred to Hogwarts through some sort of magical accident. Maybe they'd have to deal with the different magical system there compared to their own world. They could make new friends among the Hogwarts students and get involved in the typical Hogwarts adventures like the Triwizard Tournament or dealing with a new dark threat.
The main plot could revolve around fixing a mistake in the past that has led to a distortion in the present. For instance, something might have happened in the past that changed the balance between the different camps of demigods. Percy, Thalia and Nico go back to set things right. They'd encounter different cultural and technological aspects of the past time period. And they would likely face opposition from those in the past who don't understand their motives or are being influenced by the very thing they are trying to correct.
In a 'Percy and Nico are betrayed' fanfiction, perhaps they are betrayed by the gods. The gods, fearing their growing power, might spread lies about them among the demigods. Percy and Nico then have to fight not only against the false accusations but also the demigods who turn on them. They would have to find a way to clear their names and prove their loyalty.
Well, it could be that Percy yells at Nico because he feels Nico is being too reckless in a dangerous mission. Maybe Nico is about to do something that could endanger the whole group, and Percy, being the protective type, loses his temper.
In this fanfiction, perhaps Percy adopts Nico after Nico has been ostracized by the other demigods. Percy realizes that Nico needs a safe space. So he offers his home. As time passes, they start to understand each other better. Nico might open up about his past traumas to Percy, and Percy would use his own experiences to help Nico heal and feel like he belongs.
Well, it could be that Nico has some dark plan. Maybe he believes Percy is the key to saving someone he loves in the Underworld. He might think that by kidnapping Percy, he can force Percy to use his powers to bring that person back to life or something like that.
Perhaps in the fanfiction, Nico has been tricked by an enemy. This enemy convinces Nico that Percy is a threat to their world. So, Nico decides to take matters into his own hands. Another possibility is that Nico's own internal demons, related to his past traumas and his complex relationship with the gods, drive him to see Percy as an obstacle that needs to be removed.
Perhaps it's a fanfiction focused on their relationships. They might form an unexpected bond in the story. Nico, who has often been a bit of an outcast, could find real friendship and acceptance from Percy and Thalia. They could start out on a simple quest, like finding a lost artifact, and during this journey, they learn more about each other and grow closer as friends.