He could have the power of light. Maybe he could create blinding flashes to disorient his enemies. Also, he might be able to control the sun's heat, using it in combat like shooting out beams of intense heat.
If Naruto was the son of Apollo in fanfiction, he could possess the power of prophecy. Just like Apollo was associated with prophecy in Greek mythology. He could also have enhanced speed and strength, as if he was given the physical gifts of a god. Additionally, he might be able to heal himself or others quickly, which would be a very useful power in the dangerous world of Naruto.
He might have the power of instant teleportation. Just like a god can appear anywhere, Naruto could do the same to surprise his enemies or reach his friends in need quickly.
He might have the power of Odin's All - Father magic. This could allow him to see the past, present, and future, just like Odin's ability to gain wisdom through his ravens. Also, he could potentially have enhanced physical strength, much greater than what he has as a regular ninja, allowing him to overpower even the toughest opponents with ease.
He would probably have powerful water - based abilities. For example, he could control the oceans and large bodies of water like Poseidon. He might be able to create huge tidal waves or even walk on water effortlessly.
He might have the power of charm. Just like Aphrodite can make people fall in love with a glance, Naruto could have a natural charisma that makes people trust him and want to follow him more easily. It could be really useful in his mission to bring peace to the ninja world.
In this fanfiction scenario, Naruto would inherit a host of Saiyan powers on top of his ninja skills. For starters, he would have the Saiyan's natural fighting instincts and the ability to quickly recover from injuries. His power level would constantly increase as he fought, similar to how Saiyans get stronger through battles. His existing jutsus like the Rasenshuriken could be infused with Saiyan energy, making it not only cut through opponents but also blast them with a powerful ki - like force. And with his Saiyan tail, if he had one, he might have an additional form or power that was unique to him, perhaps something that combined the power of the Great Ape form with his Nine - Tailed Fox form.
He would have a whole range of powers. He could have the strength of Hercules to overpower his opponents. Also, the ability to control the wind like Aeolus from the Olympians. This would enhance his existing wind - style jutsu.
Naruto could inherit Frieza's ability to survive in space and extreme environments. Along with that, his Rasengan could take on new forms influenced by Frieza's energy. For example, it could be a purple - colored Rasengan with the destructive power of Frieza's attacks. He may also gain the power to transform like Frieza, with each transformation increasing his strength, speed, and overall combat capabilities.
He could inherit time - related powers. Maybe the ability to slow down or speed up time, which would be a huge advantage in battles. He might also be able to see into the past or future, which could help him in making strategic decisions.
He would likely have a combination of ninja powers and Asgardian powers. For example, he could still use his Rasengan, but it might be enhanced with Asgardian magic. He might also have superhuman strength and durability like the Asgardians, which would make him even more formidable in combat.
He might have the power of earth manipulation. Since Rhea is associated with the earth, Percy could gain the ability to control the ground, like creating earthquakes or forming barriers made of stone. This would be a very powerful addition to his existing powers.