Some of the gods can also be common characters. For instance, Poseidon, Percy's father. He might play a role in either helping or hindering Percy's revenge, depending on the plot. Zeus could also be involved if Percy's revenge somehow crosses paths with the affairs of the sky god. And then there are the other demigods at Camp Half - Blood who could join Percy in his quest for revenge or oppose him if they have different views.
One interesting plotline could be Percy seeking revenge on a new enemy who has wronged his friends at Camp Half - Blood. Maybe this enemy has been stealing magical artifacts from the camp, and Percy decides to use his water powers and his wits to track them down and make them pay.
Perhaps it's a gift from his father, Poseidon. Since Poseidon has control over the sea and all things related to water, he might have some connection to a power that gives Percy wings, like as a way to help Percy reach new heights both literally and metaphorically.
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.
There shouldn't be any legitimate fanfiction with such a plot. The Percy Jackson series has a certain set of rules and moral and ethical boundaries within its fictional universe. The idea of Percy getting Hera pregnant is not only out of character but also violates the general tone and values of the series.
In such fanfictions, Percy's character is often explored through his resilience. For example, if he's physically abused, we see how he still tries to fight back or find a way to escape, which shows his determination. His loyalty also comes into play as he might not want to burden his friends with what he's going through.
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.
Well, in a 'Percy Jackson Gets Arrested' fanfiction, perhaps he could be framed by a monster in human form. The police, not knowing his true identity as a demigod, might take him into custody. This could lead to a whole lot of chaos as his friends from Camp Half - Blood would surely try to break him out. They would have to figure out how to deal with the mortal legal system while also fighting off any monsters that might be taking advantage of the situation.
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.