A great plot could revolve around Percy's role in the power dynamics among the gods. As the son of Hera and Poseidon, he has the potential to shift the balance. Some gods might try to use him as a pawn, while others see him as a threat. Percy would then have to navigate this complex political situation among the gods while also dealing with his own personal growth and the challenges that come with being a demigod.
A plot idea might be that Percy is magically transported to a time when Poseidon and Hera had a major disagreement in the past. He has to use his wits and powers as Poseidon's son to prevent this rift from growing and causing chaos in the present. Also, he could be given a task by Hera to rescue one of her favored heroes who has been trapped by a sea monster under Poseidon's domain. This would put Percy in a tricky situation where he has to balance his loyalty to his father and the task given by Hera. He might also encounter a group of demigods who are descendants of both Poseidon and Hera in some way, and together they have to face a common enemy.
If Percy were the son of Poseidon and Hera, his powers would be quite unique. His water powers could be enhanced in a different way. Maybe instead of just controlling water, he could transform water into different substances related to Hera's domain, like turning water into a fertile soil - like substance for plants to grow. His leadership skills would be off the charts. He could use his Poseidon - given strength and Hera - influenced charisma to lead not just demigods, but also other mythical creatures. He might also have the power to protect a place or a group of people in a more profound way, like creating a shield made of both water and a golden energy that represents Hera's protection.
One plot idea could be that Percy has to deal with the internal conflict of having two very different divine parents. Hera's strict and often - conniving nature clashes with Poseidon's more laid - back and powerful - but - less - controlling side. Another could be that the other demigods at Camp Half - Blood are initially distrustful of him because of Hera's reputation. So Percy has to prove himself not only as a powerful demigod but also as someone who can be trusted.
It's possible in fanfiction, but it would be a huge departure from the established canon. In terms of how it would change the story, the entire plot of the series would need to be re - imagined. For example, Percy's initial discovery of his parentage would be much more complex. Instead of the straightforward revelation of being Poseidon's son, there would be the added mystery and confusion of Hera's involvement. His place at Camp Half - Blood might also be questioned more, as Hera has a different standing among the gods compared to Poseidon.
There's also a fanfic called 'Poseidon's Promise'. In this one, Percy has to deal with a mysterious prophecy related to his father. The story is full of twists and turns. It delves deeper into the relationship between Percy and Poseidon, which is really interesting to read about. It also has some great action scenes with Percy using his water - based powers.
One idea could be Percy feeling betrayed by Poseidon and then finding a new family among other outcasts in the demigod world. He could form a group with them and go on adventures that are not related to Poseidon at all.
Percy's powers would be more complex. He could have the ability to communicate with sea deities in a different way. Maybe he could use Hera's diplomatic skills to make deals with them or resolve disputes. His control over water could also be extended to other substances that have a connection to water, like steam or ice, in a more efficient way than before. And his powers of persuasion, which could be influenced by Hera, might be used to calm angry sea storms or make sea monsters cooperate with him, which is an interesting combination of his Poseidon - given powers and Hera - influenced skills.
One interesting fanfic could be where Percy has to deal with the complex politics of having two powerful parents. He might struggle to find his place among the demigods who have more traditional parentage. For example, some might be jealous of his unique heritage, while others might fear his potential power.
In fanfiction, their relationship could be developed by having Poseidon be more involved in Percy's daily life. Instead of just popping in during big events, he could be there for the small things too. For instance, helping Percy with his schoolwork that involves marine biology. They could also have a project together like building a new safe haven for sea nymphs. Through this, they would bond over creating something positive. Also, Percy could start to take on some of Poseidon's responsibilities in the ocean, with Poseidon guiding him. This would not only strengthen their bond but also show Percy growing into his role as Poseidon's son.