Percy might start to think about different types of power systems. Naruto's use of chakra is very different from Percy's powers related to the Greek gods. This could make him consider alternative ways of using and developing his own abilities.
His view of friendship could expand. In Naruto, friendship is very strong and characters are willing to sacrifice a great deal for their friends. Percy might see new ways to show loyalty to his own friends.
Percy could gain a new understanding of the power of storytelling. Seeing how fans can take his world and create new and exciting tales might make him appreciate the stories of the Greek gods more. He might also start to see his own story as part of a larger tapestry that can be woven in countless ways by different people, which could give him a more inclusive and open - minded view of his own place in the world.
His view of power might change. In The Hunger Games, the Capitol holds extreme power over the districts, which is different from the power dynamics in Percy's world. He might start to think more about how power can be misused and how the oppressed can fight back.
His view of himself might become more complex. If he reads fanfictions where he is depicted as a super - powerful hero beyond what he already is, he could start to question his own limits. Or if he reads ones where he fails a lot, it might make him more self - critical.
It could also strengthen his bond with his friends. Realizing that others are interested in their story and are creating their own versions of their relationships could make Percy feel closer to his friends. He might even share some of the interesting fanfiction ideas with them and have a good laugh together.
His view might expand. He'd realize there are countless other possibilities beyond what he knows in the ninja world. For instance, if there's a crossover with a futuristic world, he'd see how technology could be integrated into combat, which might make him think about new ways to use his own jutsu.
As a father, Naruto might project his own values onto the Percy Jackson world. His strong belief in hard work and never giving up, which he would also teach his child, could make him look at the demigods' battles and quests as opportunities for growth. He may think that the demigods, like his own child, can learn important life lessons through their experiences in the Percy Jackson universe.
His view could become more nuanced. If the fanfic shows different versions of the events in the 'House of Hades', he might start to see the original story in a new light. For example, if a fanfic focuses on a side character's perspective during the events in the 'House of Hades', Percy might realize there were more things going on than he initially thought.
It could make them more aware of how mortals misunderstand them. In fanfiction, there could be inaccuracies about the Titans. This might annoy some Titans, but it could also make them realize that mortals don't have the complete picture. So they may become more interested in clarifying their true nature to mortals if they ever interacted with the mortal world again.
It could make him appreciate the original series more. By seeing different takes on the story, he'll notice the unique aspects of the original plot and characters that the author created.
Percy Jackson's personality could change quite a bit in another world in fanfiction. His sense of humor might be affected as the new world may not have the same cultural references that he's used to in his own world. He could become more solitary as he tries to figure out this new place on his own. However, his underlying sense of justice would likely remain. He would still be drawn to help those in need, but he might go about it in different ways. He might also develop a new sense of pride in being able to adapt and survive in this new world, which could change how he presents himself to others.