A plot could be that Percy accidentally breaks an ancient and sacred rule of the camp while trying to save a fellow camper. As a result, he's kicked out. He then decides to seek refuge with his mortal family, but soon realizes that monsters are targeting them too because of his connection. So he has to protect his family while also finding a way to get back into the good graces of the camp.
Maybe there's a power struggle within the camp. A group that doesn't like Percy's growing influence manages to convince the camp council to kick him out. Percy then goes into the mortal world and starts to train on his own. He discovers new fighting techniques and magic that were unknown to the camp. Eventually, he returns to the camp stronger than ever, ready to expose the corruption within the council that led to his expulsion.
In one fanfiction idea, Percy is kicked out because a new camper arrives who has a personal vendetta against him. This camper has connections within the camp and spreads false rumors about Percy being a traitor. Once out, Percy travels to different mythological places he's heard about but never visited. He meets strange creatures and gods who help him realize that his true power doesn't come from the camp but from within himself. He then decides whether to return to the camp and prove his worth or create his own safe haven for demigods.
One reason could be if he was wrongly accused of stealing something important from the camp. Maybe a powerful magical item goes missing and Percy is the prime suspect because of his past issues with authority. Another reason might be if he accidentally endangers the camp in some way during a mission or training exercise. For example, if he unknowingly leads some dangerous monsters back to the camp.
One possible reason for Percy being kicked out in fanfiction could be that he was wrongly accused of stealing a powerful artifact from the camp. Maybe someone framed him to get him out of the way. As for what happens next, he could go on a journey to clear his name, finding allies outside the camp who believe in his innocence.
Percy could also be very sad. He has so many memories and relationships at the camp. He might go into a state of depression for a while, not knowing where to go or what to do next. But then, he would probably start to think about how to get back in or how to make things right. He might start looking for evidence to clear his name if he was wrongly accused. He could also look for new allies outside of the camp who could help him in his situation.
Perhaps Percy was kicked out because of a prophecy that the camp elders misinterpreted. They thought that Percy's presence would bring great danger to the camp, so they made the hasty decision to expel him. It could also be that there was some sort of political intrigue within the camp. Some powerful figures at the camp might have seen Percy as a threat to their power and influence, and so they fabricated reasons to get him kicked out.
He might be really angry at first. He's given so much to the camp, and being kicked out like that without a proper explanation would make him furious. He could storm off, vowing to find out the truth and clear his name.
Well, one possible plot could be Naruto leaving Konoha and finding a new group of allies who are also outcasts. They then form a powerful team to prove themselves to Konoha.
Maybe there's a prophecy at camp that some misinterpret to mean Percy will bring destruction. So, the campers, out of fear for their safety, betray Percy. They could lock him up or try to send him away, even though Percy has always been loyal to the camp. He then has to find a way to either clear his name or find a new place where he belongs.
Another plot could be Naruto's journey of self - discovery. After being kicked out, he realizes his own worth independent of his family name. He may focus on developing his skills even more, like his ninjutsu and his ability to make connections with people. He might start to understand that family doesn't have to be blood - related.
One interesting storyline could be about a new demigod arriving at Camp Jupiter and having to prove themselves in unique challenges. For example, they might have to face a long - forgotten Roman monster that has awakened near the camp. Another could be a love story between two characters from different cohorts within the camp, like a centurion falling for a new recruit. And there could also be a story where the campers have to deal with a mysterious prophecy that seems to point towards a great danger not just for the camp, but for the entire Roman demigod world.
One interesting storyline could be about a new demigod arriving at Camp Half - Blood and having to prove themselves. They might face challenges that the original characters in the Percy Jackson series didn't, like dealing with a new type of monster that is specifically targeting new recruits. Another could be a fanfiction where the gods interact more directly with the campers in ways that are not in the original books. For example, Zeus coming down to give a special training session to the lightning - wielding demigods.