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.
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.
Percy would be extremely angry. He loves his mother deeply, and her disownment would feel like a huge injustice. He'd probably want to immediately go and confront the people who disowned her. Maybe he'd also feel guilty, thinking it's his fault as a demi - god that his mother is in this situation.
In the fanfic, it could be that Percy gets Triton as a gift from Poseidon. Triton could be initially reluctant to be with Percy, seeing him as an upstart half - blood. But as they spend time together, they start to understand each other better. Triton might help Percy improve his control over water, especially in more complex and powerful ways. They could go on quests together, like finding a lost ancient sea artifact that has the power to either save or destroy the world of the sea gods.
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.
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.
One possible plotline could be that Sally is disowned by her family for being involved with the world of the gods. Percy then has to go on a journey to find a way to restore her family's acceptance. He might face various mythological creatures and challenges along the way. Another idea could be that Sally's disownment is part of a plot by an enemy to get to Percy. So Percy has to protect his mother and also figure out who is behind this scheme. It could also be that Sally discovers a dark secret about her family that leads to her being disowned, and Percy helps her uncover the truth and clear her name.
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.
You can start by looking on fanfiction websites like Archive of Our Own (AO3) or FanFiction.net. These platforms have a large collection of fanfics, and you can simply search for 'Percy Jackson disowned' in their search bars. There are also some Percy Jackson - focused fan communities where fans might share their own 'disowned' fanfics or recommend ones they've read.