Perhaps when Percy meets this half - blood at school in the fanfiction, it starts with a bit of suspicion. But as they start talking, they realize they have a connection. They could find out that they are related in some way through the complex family trees of the half - blood world. Then, they might go on a quest within the school grounds, like finding a hidden artifact that has been causing strange things to happen. This would not only strengthen their bond but also prove their worth as half - bloods to the other students and any hidden powers or entities within the school.
Well, they might become instant friends. They could discover they have similar powers or face the same challenges at school, like dealing with mean teachers who don't understand their special needs. Maybe they'll team up to fight off some bullies or figure out a mystery together.