Well, it could be that Naruto, with his ninja skills, enters the world of Percy Jackson. Maybe he teams up with Percy to fight some common enemies like the Titans or other mythological creatures. Naruto's jutsus would be a cool addition to the battles in the Percy Jackson universe.
Naruto's adaption in the Percy Jackson universe in this fanfiction could be really fascinating. His strong sense of loyalty and friendship, which are core aspects of his character in Naruto, would likely carry over. He might form alliances with the demigods in Percy Jackson. His ninjutsu could be seen as a form of unique power. For instance, his Rasengan could be a powerful attack against the monsters. Also, Naruto's growth as a character in his own world was about learning from his mistakes and becoming stronger. This same growth pattern could be applied in the Percy Jackson universe, where he starts as an outsider but gradually becomes an important part of the new world.
Maybe in the fanfiction, the Gods decide to test Naruto and Percy Jackson by pitting them against each other in a series of trials. This could explore how their different backgrounds and abilities influence their performance in the trials.
A possible plot would be that Naruto arrives during a time of unrest in the Percy Jackson world. His presence as the God of Destruction shakes up the balance of power. Some might see him as a savior, thinking his power can end the chaos, while others fear he will only make it worse. He would have to deal with the politics among the gods, like Zeus being suspicious of his power. Naruto could also encounter demigods who are both in awe and afraid of him. He might try to teach them some of his ninja ways, but this could lead to misunderstandings and more 'destruction' as they try to adapt his skills to their world.
One common theme is power. Since Naruto has his own set of powers and Percy Jackson has his demigod abilities, when combined with the idea of a primordial god, power dynamics are often explored.
In a 'Percy Jackson, Naruto, The Forgotten God' fanfiction, perhaps the forgotten god has some power that is crucial for both their worlds. Percy Jackson might come from his world where Greek gods exist and Naruto from his ninja - filled land. They are brought together by some prophecy. They start their journey, facing enemies from both their universes who don't want the forgotten god to be found. Along the way, they learn from each other's fighting styles and ways of life.
He would likely have enhanced versions of his existing abilities. His speed, which was already remarkable as a ninja, would be god - like. He could move so fast that he would seem to be in multiple places at once, much like the gods' ability to teleport or move at incredible speeds. His strength would also be magnified, allowing him to lift mountains or large structures with ease.
Well, one way could be that in the fanfiction, Naruto's powers are somehow related to the Shinto gods. Maybe Naruto is on a quest in the Percy Jackson universe and he encounters a Shinto god who bestows him with a new power or task. For example, a Shinto god could give Naruto the ability to control water in a more magical way, similar to how demigods in Percy Jackson have special powers related to the gods.
Poseidon is a likely candidate. Since he is Percy's father in the original series, it would make sense for him to take a more active role in raising Percy. He could teach Percy all about the ocean, its secrets, and how to use his water - related powers more effectively.