Well, a plot could involve Percy being cursed with uncontrollable godly strength for a while. He has to figure out how to manage it while also dealing with normal demigod problems like school and relationships. Maybe there's a new prophecy that only someone with his level of strength can fulfill, but it requires him to make a great sacrifice. Or, he could be in a situation where he has to use his strength to hold up a collapsing magical structure that is protecting a lot of innocent people.
One idea could be Percy having to face an ancient and powerful monster that was once sealed away by the gods. His godly strength is put to the test as he discovers new abilities he didn't know he had. Another could be Percy competing in a special Olympics - like event among demigods where his godly strength gives him an edge, but also makes him a target. And perhaps he has to use his strength to save a group of mortals who accidentally wandered into a dangerous demigod territory.
A great plot for this fanfic could be that Percy's godly strength starts to attract the attention of some of the more jealous and power - hungry gods. They start to plot against him, trying to either take his strength or get him out of the way. Meanwhile, Percy is using his strength to help his friends in a series of adventures. For example, he might use it to lift a huge boulder that is blocking the entrance to a secret place where they can find a powerful weapon to defend themselves against an upcoming threat. Also, his strength could play a key role in rescuing a kidnapped demigod friend, as he battles through hordes of monsters with his enhanced power.