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.
In a 'Percy Jackson leaves earth' fanfic, he might end up in a whole new realm of gods and monsters. Maybe he discovers a parallel universe where the rules of the Greek gods are different. He could encounter new versions of his friends or enemies and have to learn new powers to survive.
Perhaps he'll meet other demigods who have also left Earth for various reasons. They could form a new group and face new challenges together. For example, they might have to deal with a powerful alien being who wants to use their demigod powers for its own evil purposes. Percy would then need to use his wits and strength to lead the group and find a way to defeat this new enemy.
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.
Percy might be very confused at first. He would wonder why his mother, who has always been there for him, suddenly left. He could start to question if he did something wrong. In some fanfics, he may go into a state of denial, not wanting to believe that she's really gone.
Another reason might be that she is forced to leave. There could be a magical or divine force at play. Maybe a god or goddess, who has some grudge against Percy or his father Poseidon, manipulates the situation so that Sally has no choice but to abandon Percy. This would add a lot of drama and conflict to the fanfic.
There could be a fanfic where Percy leaves because he feels out of place after all the battles. He decides to start a new life away from the demigod world. In the fanfic, he tries to adapt to a normal human life but finds it difficult as his demigod instincts keep getting in the way.
Some minor gods related to the earth like Demeter could be allies. She has her own powers over the earth and could teach Percy some of her techniques. Also, the nymphs, such as the water nymphs and the tree nymphs, could be allies. The water nymphs could help with the water sources within the earth (like underground rivers) and the tree nymphs could help Percy take care of the forests. And let's not forget the pegasi. They could be used for transportation when Percy needs to quickly travel across his earthly domain.
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.
He might go on a solo adventure to discover more about his powers. He could encounter new mythical creatures that he hadn't seen around the camp. Maybe he'll meet some other demigods who are on the run or have their own secret quests.