In the fanfiction, Percy's relationship with Zeus may develop gradually. Zeus, being a powerful and often aloof figure, might initially be distant from Percy. But as Percy shows his courage and abilities, Zeus could start to take a closer interest. This might lead to Zeus mentoring Percy in some ways, teaching him about godly powers and responsibilities.
There is definitely some out there. One aspect that might be interesting in these fanfictions is how Percy's powers could be different since his parentage is from Zeus and Hera. In many traditional stories, Percy is a son of Poseidon, but in these fanfictions, he could have lightning - based powers or other abilities related to Zeus and Hera's domains.
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.
One possible aspect in this fanfiction could be Percy's struggle with his new identity. As the adopted son of Zeus, he might face challenges in a world where the relationships between gods and their offspring are complex. He could be dealing with the expectations of Zeus, which are likely to be high and different from what he has known before.
To write a compelling Percy, give him flaws. Maybe he's overly impulsive because of his divine heritage, which often gets him into trouble. Also, show his internal struggle with being the son of Zeus. He could be constantly torn between living up to his father's expectations and forging his own path.
One common theme is the struggle for identity. Percy, who is usually associated with Poseidon, now has to come to terms with being the son of Zeus and Hera. Another theme could be power - how he manages and discovers his new powers.
A potential plot could be Percy being sent on a special mission by Zeus. This mission might be crucial for the balance of the godly world, and Percy has to prove himself worthy of being Zeus' adopted son. He could face many obstacles and powerful foes during this mission.
Yes, there are quite a few interesting ones. You can start by looking on fanfiction platforms like Archive of Our Own. Some fanfics might explore Percy's relationship with his divine parents in unique ways, or perhaps put him in new and exciting adventures that are different from the original series.
Percy would probably become more arrogant at first. But as he faces challenges related to his new identity, he'd learn humility. His loyalty to his friends might be tested as Zeus could demand things that conflict with his friends' interests. He'd also have to deal with the expectations of being the son of the king of the gods, which would shape his character in ways different from when he was just Poseidon's son.
He could have the power of lightning, like Zeus. Maybe he can control it better than other demigods. Also, he might have enhanced strength and speed, similar to Zeus's own capabilities.