Perhaps Percy gets sick because of a curse placed on him by an enemy. His sickness makes him vulnerable not only physically but also mentally. His friends notice a change in his behavior. Instead of being the brave and confident Percy they know, he becomes withdrawn and moody. They start a quest to find the source of the curse and break it. Along the way, they face many challenges, and Percy has to fight his inner demons as well as his physical illness.
Well, one possibility is that Percy's friends try to find a cure. Maybe Annabeth uses her wisdom to research in the library of Camp Half - Blood while Grover tries to use his nature - related powers to find some special herbs. And Percy is just lying in his cabin, feeling really weak and feverish.
In a 'Percy Gets Sick' fanfiction, it could start with Percy getting sick after a battle with some new and strange monster. His demigod powers might be affected, so he can't even control water like he usually does. His friends are worried sick. They call Chiron for help, and Chiron examines Percy, trying to figure out what's wrong. Meanwhile, Percy has some strange dreams related to his sickness, which could be a sign of something more sinister.