Another reason could be to give Percy a chance to find himself. In the original stories, he is often defined by his role as a demigod and his relationships. When he leaves, he can start to figure out who he is without those labels, and authors can use this to delve deeper into his personality and motives.
To add drama. When Percy faints, it creates a moment of high tension. His friends or allies are then left to deal with the situation, which can lead to all sorts of interesting plot developments.
Some people write it because they might not like Percy's character arc in the original books. They feel that he gets too much praise or that his development isn't as good as it could be.
People write 'Percy Jackson tortured fanfiction' to explore darker themes. They want to see how the characters would react under extreme pressure. Percy is a strong character in the original series, and by torturing him in fanfiction, writers can show his limits and how he might overcome or succumb to different forms of torture.
It's highly unlikely that there is a legitimate 'Percy Jackson' fanfiction with such a plot. The 'Percy Jackson' series is aimed at a relatively young audience, and this kind of plot is not only against the moral fabric of the series but also goes against the established mythology. In the Greek and Roman mythologies that the series is based on, Juno/Hera has a very specific role and set of relationships, and this scenario would be a huge deviation.
People write it because they love the relationship between Annabeth and Percy. Their connection in the original series is so strong that fans want to explore more romantic moments between them, like their kisses.
It's mainly because fans adore Percy and Annabeth as a couple. Writing about their kiss allows fans to add their own spin on the relationship. They can create different scenarios, like a first kiss in a new and unexpected location, or a kiss that saves the day in a unique way. It gives fans a chance to be part of the story's romantic development.
He might meet other inter - galactic beings. For example, some sort of space - faring demigods from a far - off galaxy. Or he could encounter powerful entities that are the rulers of other planets.
He could go to New York City. It has a lot of water sources like the Hudson River where he can use his powers. Also, there are many places to hide from monsters.
He could go to the Underworld of another planet, if such a place exists in the universe he lands in. It might be filled with different kinds of dead souls and ruled by a different kind of Hades - like figure.
One possible reason could be copyright issues. The creators or rights holders of Percy Jackson might be strict about fanfiction to protect their intellectual property. They may not want unauthorized stories to be created using their characters and settings.