It could be that Percy's relationship with a certain god or goddess goes horribly wrong. Say he offends Aphrodite in a way that she convinces the other gods to disown him. Percy then has to struggle not only with the loss of his divine family but also with the isolation from the demigod community that follows such a disownment.
Well, it could be that Percy has done something that goes against his family's values or rules. Maybe he's made a choice in his relationships or his actions that his family deems unacceptable, like befriending the wrong people or getting involved in a dangerous or taboo activity. And as a result, they disown him, leaving him to fend for himself and figure out his new place in the world.
It could be a jealous demigod. Maybe someone who has always been envious of Percy's popularity and power. This demigod could spread lies about Percy that lead to his disownment, making them the main antagonist.
One possible scenario could be Percy being disowned because of his powers being seen as too dangerous. Maybe the other demigods or even his own family in the fanfic fear what he can do and decide to cut ties.
In this fanfiction, the plot could start with Zoe being cast out by Poseidon. Percy, being the hero he is, steps in. He first has to find Zoe as she might be in hiding or on the run. Then, he would have to fight off any pursuers. Along the way, they might develop a deeper bond as Percy helps Zoe come to terms with her new situation. Maybe they even team up to prove to Poseidon that Zoe is worthy.
Well, perhaps he could die sacrificing himself to save his friends from some ultimate evil. Then, his rebirth could be linked to a powerful ancient magic. Maybe a forgotten god decides to bring him back for some grand purpose, like to stop a new threat emerging in the world of the gods.
There could be a rival within the family. Someone who wants Percy out of the way, perhaps for inheritance reasons or because they're jealous of Percy's abilities or popularity. This rival could be the one who instigates the events that lead to Percy's disownment. And then there might be a mentor - like figure outside of the family who takes Percy in after he's disowned and helps him to grow and face his new situation.
He might become an even more powerful figure. The trident could enhance his abilities to control water and perhaps give him new powers related to the sea. He could use it to defend his friends and fight against stronger enemies.
Perhaps he gets the power of astral projection. He could send his spirit out of his body to explore areas that are otherwise inaccessible or to spy on his enemies. This would be a very useful power especially when dealing with secretive foes or trying to find hidden places in the mythological world.
One possibility in this fanfiction is that the wings change Percy in some way. They might give him a new perspective, literally and figuratively. As he soars through the skies, he could see things he's never noticed before on the ground. This could lead to him discovering new secrets about the demigod world or even his own powers. He might also face new challenges, like dealing with air - based monsters or learning to fly in different weather conditions.
Well, I'm not sure of the exact titles, but a good 'Percy Jackson disowned' fanfic would probably explore Percy's emotions deeply. It could show him feeling betrayed by his friends and the gods. Maybe it would have him going on a solo adventure to prove his worth. He might encounter new allies and enemies in the process. It could also be interesting if the fanfic explored how the other demigods reacted to his disownment, whether they supported him or not.