One way it could work is if Naruto's immortality was a result of some powerful magic in the Percy Jackson universe. Maybe he was blessed by a god or came into contact with a magical artifact that made him immortal. His ninja skills would be quite unique in that world and could make for some interesting storylines, like teaching demigods his stealth techniques.
Well, he could potentially use his ninja skills to help the demigods in various quests. Maybe his shadow clone jutsu could be used for scouting or distraction, which would be a huge advantage against the monsters in the Percy Jackson world.
In a crossover fanfiction, the settings could merge. For example, Camp Half - Blood could be visited by Naruto. The demigods would be intrigued by his chakra abilities, which are so different from their own powers based on the Greek gods. Naruto could teach them some of his ninja skills in return for learning about their combat techniques. And perhaps the Juubi's presence could disrupt the natural balance of the Percy Jackson world, leading to new and unexpected adventures for both Naruto and the demigods.
Well, one way it could work is by having Naruto possess the speed and cunning associated with Hermes. Maybe he could use his ninja skills in the world of Percy Jackson. For example, his shadow clone jutsu could be a really unique ability there. He could send clones to gather information quickly, just like Hermes' messengers.
One way it could work is by having Naruto's powerful chakra be misinterpreted as the kind of godly power in the Percy Jackson universe. Maybe he arrives in that world accidentally through some sort of interdimensional rift. His ninja skills, like his shadow clone jutsu, could be seen as some strange and powerful magic there. And his determination and willpower, which are so central to his character in Naruto, could be what makes him seem like a 'god of destruction' in the context of Percy Jackson's world where the gods often have very strong wills and tempers.
In a Percy Jackson fanfiction where Naruto is the son of Susanoo, Naruto's ninja - like agility and stealth would be a great asset. Susanoo in Japanese mythology is a powerful entity, and if Naruto is his son in this fanfic, he could have some sort of control over powerful weapons or shields. He could use these in battles against the Titans or other enemies in the Percy Jackson world. Also, his ability to never give up, which is a big part of Naruto's character, would make him a formidable hero in this new setting.
If Naruto was the son of Zeus in a Percy Jackson fanfiction, his character would have to adapt to a whole new world. The demigod world of Percy Jackson has its own rules and politics. Naruto, being used to the Hidden Leaf Village's system, would have to learn about cabins, quests, and the whole Greek mythology aspect. His shadow clones could be used in really creative ways during quests, like scouting ahead or fighting multiple monsters at once. And his ability to never give up would be a great asset when facing the many challenges that come with being a demigod.
His role might be that of a protector. Naruto is known for his selfless nature. In the Percy Jackson world, he could use his immortality to constantly safeguard the demigods and the places they hold dear, like Camp Half - Blood. He could also be a bridge between the ninja world he comes from and the world of demigods, bringing in new techniques and ideas.
He could be a mentor figure. Given his immortality and vast combat experience from the Naruto world, he could teach the demigods new ways to fight and strategize.
Well, in this fanfiction, Percy's immortality could lead to some really interesting character development. He may start to feel lonely as he realizes that he will outlive all of his friends. This could cause him to be more withdrawn at first. But then, as he interacts with the students at Westover Hall, he might find new meaning in his immortality. He could start to see it as a way to protect others not just for a short time, but for as long as he exists. He could also form deep and lasting friendships with those who accept him for who he is, regardless of his immortality.
One of the things that makes Naruto and Percy Jackson NPU fanfiction exciting is the potential for character development. For instance, Percy could learn from Naruto's experience of being an outcast in his village at first and then rising to become a hero. Naruto, in turn, could gain a better understanding of the complex family dynamics that Percy has with his godly parent and his mortal parent. Their friendship could be a catalyst for them to overcome their own personal struggles in new and interesting ways.