The kid could be a young demigod whose parents are unknown. Maybe he's been living on the streets, hiding from monsters, until Percy finds him.
Perhaps the kid is a mortal who has somehow become involved in the demigod world. He could have witnessed something he shouldn't have, like a monster attack, and Percy feels responsible for protecting him. So, Percy decides to adopt the kid and keep him safe from the dangers of the demigod world that he's now been exposed to.
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.
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.
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.
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.
The kid could have the ability to communicate with animals like Artemis, but in a more water - influenced way. For example, they could communicate with sea creatures as well as land animals. This would give them an edge in various situations, like getting help from dolphins during a sea battle or having wolves guide them through a forest.
The story could explore the challenges of raising a child in a world where the gods have their own rules. Percy and Artemis would face opposition from other deities. The kid could grow up being an outcast among the gods but also having amazing potential. It might be about the kid finding their place in the world and the sacrifices Percy and Artemis would make for their child.
It's possible that the kid is a result of some sort of time - travel or alternate reality situation. Maybe the kid is actually Naruto from a different timeline or dimension, and Naruto has to figure out how to deal with this strange situation while on his training trip.
Sally Jackson, Percy's mother, is also an important character. Her love for Percy and her strength can be a central part of the story. She could be the one who gives Percy the emotional support he needs when he's confused or scared about the strange things happening around him. In a fanfiction, she could also have some secrets of her own related to the mythological world that Percy discovers as he grows up.
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.