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.
There's a chance he'd end up in a world made entirely of water. Since Percy has a special connection with water, he might initially think it's a great place. But then he discovers that there are dangerous water - based creatures that are far more powerful than anything he's faced on Earth, and he has to find a way to co - exist or fight them.
Maybe he'll be transported to a realm where the laws of physics are completely different. Gravity could work in strange ways, and time might not be linear. Percy would have to adapt quickly to these new conditions while also trying to figure out how to get back to Earth or at least communicate with his friends and family there.
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.
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.
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.
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 might go to his mother's place. It's a place where he has always felt safe and loved. His mother has always been his anchor, and even though it might be a risky move as monsters could track him there, it could be his first instinct. Another option could be to go to a big city like New York. There are so many people and places to hide. He could blend in with the crowds and use the chaos of the city to stay unnoticed. He could also find refuge in an abandoned building or a place that has some sort of magical protection that he discovers by accident.
Percy might meet Sherlock Holmes. Sherlock would be intrigued by Percy's world of gods and monsters. Percy could ask Sherlock for help in solving a mystery in the mortal world that is somehow connected to the world of the demigods. And Sherlock could use his deductive skills to figure out some secrets about the demigod world.
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.
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.
In a 'Guardian of the Earth Percy Jackson Fanfiction', it could be that Percy is chosen by Gaia herself (but in a good way this time). Gaia has realized that the earth is in danger from some new threat that the gods can't handle. Percy, as a half - blood, has a unique connection to both the mortal and divine worlds. So he has to journey to different parts of the earth, from the deepest jungles to the highest mountains, to gather the elements of earth's power and use them to create a shield around the planet to keep out the dark forces that are approaching.
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.