She could be banished due to her unorthodox powers. In many fictional settings, if a character has powers that are different or not understood by the majority, they are often feared. If female Percy has some unique abilities that are considered dangerous or abnormal, the authorities or those in charge might decide to banish her to keep the peace or protect their own interests.
Well, perhaps in the fanfic, there's a prophecy that gets misinterpreted. The gods think Percy is going to cause great harm based on this wrong interpretation, so they send him to Tartarus. It could also be that there's some sort of power struggle among the gods. Percy might be caught in the middle of it, and one side decides to get rid of him by banishing him to Tartarus to tip the balance in their favor.
In the context of this fanfiction, Naruto's family might banish him because they don't fully understand his 'godlike' powers. They see the surface level of his new strength and assume the worst. Maybe they think he has been corrupted by some unknown force. Also, it could be related to power politics within the family. Some members might see Naruto's power as a threat to their own position and influence, and so they convince the rest of the family to banish him to safeguard their own interests.
Perhaps a prophecy misinterpretation. If a prophecy was misread and it seemed like Percy was the cause of great evil to come, the gods might take drastic action. For instance, if the prophecy spoke of a 'child of the sea' causing destruction, and they wrongly assumed it was Percy without fully understanding the context.
One possible reason could be that they are wrongly accused of a serious crime within the wizarding world. Maybe some dark magic items are found in their possession and the authorities suspect them of being involved with dark wizards, without fully investigating the real source of those items.
Well, it could be related to Percy's past actions. If he has a history of being unreliable in the eyes of the character rejecting him, that would be a valid reason. For instance, if he has broken promises in the past. Or perhaps there are external factors at play. Maybe the family of the person rejecting Percy doesn't approve of him, so they influence the decision. Another aspect could be that Percy is too different in terms of values. If he values adventure and risk - taking above all else and the other character values stability and safety, it can lead to a rejection.
Family drama. The Weasley family has its fair share of chaos. His relationship with his family members, especially after some of the disagreements he had, could be a source of emotional stress that manifests as migraines. For example, his estrangement from his family during a certain period in the books could have taken a toll on his mental well - being.
Perhaps Draco has been bullying some of Percy's friends or younger students that Percy feels responsible for. In fanfiction, the relationships can be tweaked a bit. Percy might be more hot - headed in some versions and not put up with Draco's pure - blood supremacist attitude towards those he deems as 'lesser' students. So he sends the Howler as a strong warning.
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.
Annabeth is a very determined and strong - willed character. In her mind, Percy is part of her 'territory' because she has fought beside him and for him. She may feel that she has a claim on him, and this sense of ownership makes her territorial.
One possible reason could be that she is wrongly accused of a serious crime she didn't commit. For example, if there was a major theft in the village and false evidence was planted against her. Another reason might be that she has some new and powerful abilities that the village elders fear. They may think she could become a threat in the future, so they banish her to protect what they see as the village's stability.