One reason could be that he values his humanity. He has friends and a family among mortals, and he doesn't want to lose those connections. Becoming a god would likely put a strain on those relationships.
One possible reason could be that Percy values his humanity. He has seen the flaws of the gods and doesn't want to be like them. In the original series, he has had many experiences with the gods' self - centeredness. So in fanfiction, he might turn down godhood to stay true to his mortal self.
One possibility could be that Percy, as a new god, would have to deal with the complex politics of Olympus. He's always been an outsider, and now he'd be thrust into a world of power plays and ancient rivalries. Maybe he'd try to change some of the archaic rules that the gods have held onto for centuries.
One idea could be that Percy turns into a dragon to fight a powerful sea monster. Since he has his connection to water, as a dragon he could breathe a sort of water - based fire or ice that freezes the water around the monster, trapping it.
His friends might be relieved. They value Percy as he is, a demigod. If he became a god, he might change, and they could lose the Percy they know. For example, Annabeth might be happy that he stays a demigod because she loves him for who he is now.
His decision might give hope to other demigods. They see that it's possible to be content without striving for godhood. For example, some demigods who might have been jealous or envious of Percy could now see that there are other paths to a fulfilling life.
Perhaps he's seen the downsides of being a god in the Percy Jackson universe. Gods are often restricted by their domains and have a lot of responsibilities. Percy, who has always been a bit of a rebel, might not want to be tied down by such rules. He likes his freedom to make his own choices and help people in his own way, not in the way a god is expected to.
Perhaps Percy turns into a falcon. In this form, he could soar through the skies and spy on his enemies from above. He might use his falcon eyes to find a secret lair of a villain or to track the movements of the Titans. It would also give him a different perspective on the world around him, seeing things from a bird's - eye view.
In a 'Percy Jackson turns evil' fanfiction, he might use his powers against his former friends. For example, his control over water could be used to flood Camp Half - Blood. His knowledge of the demigod world would also be a weapon in his hands to outwit and defeat those who oppose him.
Well, in the fanfiction, if Percy turns into a baby, he would probably lose all his skills and knowledge at first. But he might still have some instinctual connection to water. Maybe his friends like Annabeth would have to take care of him. She'd be really protective and try to find a way to turn him back.