In some fanfictions, he might be exiled because of false accusations. Maybe someone frames him for a crime he didn't commit, like stealing a powerful artifact from the camp's vault. The camp leaders, believing the false evidence, have no choice but to exile him.
He could be angry. Percy has always been loyal to the camp, and being exiled unjustly would make him furious. He might vow to prove his innocence and show the camp they made a big mistake.
One possible reason could be that he feels misunderstood. In the camp, everyone has high expectations of him as a son of Poseidon. If he makes a small mistake, it might be blown out of proportion. For example, if he accidentally causes a small flood during a training session, instead of being understood as still learning control over his powers, he could be harshly criticized. Another reason might be that he's tired of the constant danger at the camp. Every day there seems to be a new threat looming, whether it's a monster attack or some sort of internal camp conflict. He might just want to get away and have some peace for a while.
In some fanfictions, it could be that Percy has new, uncontrollable powers that the camp doesn't understand. They see him as a threat and banish him. For example, if he starts to show signs of a power that was once associated with an enemy of the camp. It's also possible that Percy's loyalty is questioned. If he befriends someone from an opposing group, the camp might think he has turned against them and banish him.
He might leave because of a personal conflict. Perhaps he has a falling - out with some of the other campers or with the camp leaders. Like if he disagrees with a new rule or policy that the camp has implemented which he thinks is unjust.
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.
In some fanfictions, Percy might be banished because he goes against the traditional ways of the demigod society too much. For example, if he tries to form an alliance with a group that the other demigods consider enemies. His actions could be seen as a threat to the stability of their world, so they banish him to keep the peace.
One possible reason could be that he is wrongly accused of a serious crime against the village. For example, if some important secrets are leaked and the village leaders suspect Naruto without proper investigation, they might exile him to safeguard the village's so - called security.
Well, one reason could be that he's wrongly accused of a major crime against the gods. Maybe some false evidence is planted, making the gods believe he's a threat, so they banish him. Another reason might be that he's trying to protect someone or something, and his actions are misinterpreted as going against the gods' will. For example, if he's trying to save a powerful but 'forbidden' creature, the gods might see this as a violation and send him to Tartarus.
One reason could be that he feels a sense of being out of place. Maybe new campers have come in and he no longer feels as important or needed. He might also leave in search of a greater purpose. In some fanfictions, he could be on a quest to find something that he believes is crucial for the safety of the camp, but he has to leave the camp itself to do it.
It could be due to a conflict with a particular authority figure at the camp. If he has a major disagreement with a camp leader over how to handle a situation, like dealing with a new enemy. Percy, being headstrong, might decide to run away and handle it his own way instead of following the camp's rules.