In this kind of fanfiction, Percy might challenge Zeus over issues like the unfair treatment of demigods. For example, Zeus could be overly harsh on demigods who make minor mistakes while the gods themselves get away with much worse. Percy, being a hero, would stand up for his fellow demigods.
Another possibility is that Percy discovers some long - forgotten weakness of Zeus. In the world of fanfiction, the author could create a scenario where Percy stumbles upon an ancient prophecy or a hidden secret about Zeus. This knowledge could give Percy the upper hand. He could then use his wits and his powers to exploit this weakness and defeat Zeus.
One interesting fanfiction could be about Percy Jackson's adventures that are more closely tied to his relation as Zeus's grandson. Maybe he discovers some hidden powers that are passed down only through the Zeus lineage. He could face challenges where his Zeus - related powers are both a blessing and a curse, like when he has to control excessive lightning powers during a battle with a powerful monster.
One possible plot twist could be that Percy, with Zeus as his father, discovers he has the power to control the skies in a way that even Zeus is afraid of. This newfound power could put him at odds with the other gods who see him as a threat to the established order.
One idea could be Percy having to mediate between Zeus and Hades during a major feud. Since he's the son of both in this fanfiction, he has a unique connection to them. He could use his powers and his knowledge of both of their domains to try and broker peace.
For the demigods, things could change a lot too. Percy is a champion of the demigods, and his victory over Zeus might mean better treatment for demigods. They could be more respected and have more rights in the world of gods and mortals. However, it could also make them targets of those who are against Percy's actions.
In some fanfictions, Zeus interacts with Percy in a hostile way at first. He sees Percy as a threat to his power and the order of Olympus. For example, he might try to limit Percy's abilities or send challenges to prove Percy's unworthiness.
One great fanfiction idea could be a story where Zeus, Poseidon and Hades have to team up to face a common threat that is even more powerful than them. Maybe a primordial being awakens and they must put aside their differences. In this story, Percy could be the mediator between them, using his unique position as the son of Poseidon to try and keep the peace among the three powerful gods.
If Zeus was Percy's father in a fanfiction, Percy's relationships with other demigods would change drastically. Percy has always been a bit of an outlier among the demigods, but this would take it to a whole new level. Those who were his friends before might start to question his motives. They might wonder if Percy will start to side more with the gods, especially Zeus, and abandon the demigods' cause. Percy would also have to deal with the fact that some demigods might try to use him as a way to get to Zeus. For example, they might ask him to put in a good word for them or try to get information from him about Zeus's plans. This could put a strain on his relationships as he has to figure out who he can trust among the demigods while also dealing with his new identity as Zeus's son.
Well, at first, they might be cautious around each other. Percy is used to being the main hero, and the daughter of Zeus's sudden appearance could make him a bit jealous or at least wary. But as they face challenges together, they start to respect each other. Maybe she saves Percy from a difficult situation, and he realizes she's not just there to steal his thunder. Over time, they could become good friends, relying on each other's unique skills in battles against monsters.