Maybe the sister had some dark secrets that she was afraid Percy would find out. For example, she could be involved in something illegal or dangerous, and to protect Percy from getting dragged into it, she chose to abandon him.
Well, it could make Percy grow up really fast. He might have to learn to fend for himself in ways he never thought before. He could become cynical about family relationships. Maybe he would start to view the world as a place where people are bound to leave you, and this could shape his interactions with others in a negative way, like always keeping people at arm's length.
Percy would likely be very confused at first. He'd probably feel a mix of anger and sadness. He might start to question his own actions and wonder if he did something wrong to make his sister leave.
One possible plot could be Percy feeling a deep sense of betrayal and loneliness. He might struggle with his self - worth as he wonders why his sister left him. This could lead him on a journey to find answers about his family and his own identity, perhaps even uncovering secrets about his demigod heritage that his sister knew and he didn't.
One possible reason could be jealousy. Maybe his sister is jealous of Percy's special abilities or the attention he gets. Another reason might be that she is forced to abandon him due to some external threat or danger that she believes will harm her if she stays with Percy. Or perhaps she has been brainwashed into thinking that Percy is a bad influence or a danger to her.
Percy might be initially very confused and hurt. He may try to find his sister and understand why she has left him. He could also become very angry and feel betrayed, which might lead him to take some rash actions.
In this fanfic scenario, Naruto's character could take a darker turn. Maybe he becomes more solitary and distrustful. His sister, on the other hand, might be favored by Kushina but have her own internal struggles. The story could explore how Naruto overcomes his feelings of being abandoned and reconciles with his family.
In a 'Naruto abandoned for his sister' fanfic, perhaps there's a prophecy that only one of them can bring peace to the village. The elders, misinterpreting it, choose the sister and abandon Naruto. Naruto then goes on a journey outside the village, meets new friends and allies, and discovers new powers. Meanwhile, the sister is having her own struggles within the village, like dealing with the pressure of being the 'chosen one' and realizing that Naruto's abandonment might not have been fair.
It could be a new powerful monster that was sent after Percy because his mom was no longer there to protect him. Maybe a monster that has a grudge against Percy's family.
Well, in such a fanfic, perhaps Percy ends up in a new camp or place full of demigods he doesn't know. He has to prove himself without his mom's support. He could befriend some unlikely characters who help him understand more about his mother's disappearance. And in the end, he might find that his mom left to protect him from a greater threat that he now has to face on his own.