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.
Perhaps the kid is from a foreign land that Naruto visited during one of his missions. The kid could be an outcast in his own place because of some unique trait. Naruto, being the kind - hearted person he is, decides to adopt him. It could be that the kid has a different kind of chakra or a strange mark on his body that makes others afraid of him, but Naruto doesn't care.
The kid could be a lost descendant of one of the old wizarding families. Maybe a long - lost relative of the Weasleys or the Blacks. Harry discovers this connection and decides to adopt the kid to preserve the family's legacy.
In this fanfiction, maybe the kid has some special power that Naruto notices. So Naruto adopts the kid to protect him from those who might misuse his power. Along the way, Naruto helps the kid come to terms with his power and they form a really strong bond, facing all kinds of challenges from other villages and ninja clans.
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.
Well, it could be that the kid has some special powers just like Harry. Harry sees himself in the kid and decides to take him under his wing. Maybe the kid is an orphan too, and Harry wants to give him a better life at Hogwarts, away from the cruel muggle world that didn't understand him when he was a kid.
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.
She could be a completely new character created just for this fanfiction. Maybe she's a normal human girl who Percy discovers has a connection to the world of the gods in some way.
The plot might involve Percy's journey in becoming a father figure. He would have to learn how to take care of a child, which is completely different from his usual adventures. He might face challenges like dealing with the girl's school, her making friends, and also protecting her from any threats that come his way because of his own past or the girl's special circumstances. For example, if she's a demigod too, there could be monsters after her.