Perhaps it's an old acquaintance from Percy's past who he reconnects with after leaving camp. They fall in love and she becomes the mother of his daughter.
Maybe Percy leaves camp to protect his loved ones. He meets someone special during his journey and they have a daughter. He then has to raise her while also dealing with the threats from his past life at the camp.
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.
He might meet other demigods who are not affiliated with the camp. These could be demigods who have been living on their own, trying to hide from the monsters and the chaos of the demigod world. They could have different views on how to use their powers and might teach Percy new ways to fight or survive.
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.
Well, it could be that Percy leaves camp and discovers a new power within himself while on his journey. He then uses this power in a very public way, like in a big sports event where he showcases superhuman abilities that no one has ever seen before. This could lead to him becoming famous as people are always fascinated by the extraordinary.
Obviously, Percy himself is the main character. There could also be his friends from the camp who might play a role in his journey outside the camp, either helping him or trying to bring him back. Maybe some new characters that he meets on his path to fame, like a mentor figure who guides him in the new world he enters.
One possibility could be that Percy leaves camp to go on a solo quest to find a long - lost magical artifact that he believes can save the camp from an upcoming threat. He might face many challenges on his own, like battling new monsters and making difficult decisions without the support of his friends at camp.
Percy would use his combat skills. He has a lot of experience in fighting monsters, so if any threats come near his daughter, he'd be ready to fight them off with his sword. He'd also use his strategic thinking to outwit enemies.
There would also be those who are angry. Some of the campers who have always been a bit jealous of Percy's status might think he's just being selfish or irresponsible. But deep down, they'd also be a bit concerned for his well - being, even if they don't show it at first.