Well, the Rinnegan in this fanfiction might give Naruto the power of the Deva Path, allowing him to manipulate gravitational and repulsive forces. Since he's greedy, he could use this to his advantage in battles, pushing away opponents and stealing their resources. He might also get the Animal Path ability, summoning powerful beasts not just for protection but to do his bidding in seizing valuable things.
He could gain the ability to absorb chakra on a much larger scale. With his greedy nature, he might use this to hoard chakra from others, making himself super - powerful. He could also gain the power to control the six paths, which he might use to dominate others rather than for the common good.
The Rinnegan could endow Naruto with the Asura Path's ability to create mechanical limbs and weapons. Given his greed, he would likely use these to amass more power and wealth. He might also gain the power of the Preta Path, which can absorb ninjutsu. In his greed, he would use this to absorb all kinds of jutsu from other ninjas, making himself the most powerful and feared ninja, all for his own selfish gains.
He might gain the ability to see chakra in a more detailed way. Since the Rinnegan is known for its enhanced visual perception of chakra, Naruto could use this to his advantage when dealing with Kurama's chakra. He could also potentially use the Rinnegan's power to enter Kurama's mindscape in a new way, maybe with more control and understanding. Another possible ability is to transfer Kurama's chakra to others in a more efficient and controlled manner using the powers of the Rinnegan.
Naruto would likely gain the ability to use all six paths techniques. The Rinnegan allows the user to manipulate gravity, as seen when Madara used it. Naruto could use this to enhance his combat skills, for example, pulling enemies towards him or pushing them away forcefully. He could also use the Preta Path to absorb chakra, which would be a great addition to his already large chakra reserves.
He could gain the ability to see different timelines clearly. The Rinnegan already gives enhanced perception, and with time travel, Naruto could view the consequences of different actions in different timelines. He might also be able to transfer knowledge from one timeline to another. For example, he could learn a powerful jutsu in the past timeline and bring that knowledge back to his original timeline.
He could have the ability to use all six paths techniques right from the start. This means he could manipulate gravity like the Deva Path, absorb chakra like the Preta Path, and even revive the dead like the Outer Path.
He could develop the ability to use the Six Paths Sage Mode more effectively. Since the Rinnegan is related to the Sage of Six Paths, it would enhance his connection to that power. He might also gain the ability to create chakra chains similar to those of the Uzumaki clan but with an added Rinnegan - powered twist.
Naruto could have the power of solar - powered regeneration. The sun is often associated with life and renewal. So, if he gets injured, his wounds could heal faster with the energy of the sun. He may also be able to use the sun's energy to enhance his physical strength. He could run faster, jump higher, and hit harder as if he was charged up by the sun's power.
Maybe he would gain the ability to control the life force of others in a new way. With the red Rinnegan, Naruto could potentially heal or harm on a much greater scale. He could revive the recently deceased by restoring their life force, or he could drain the life force of his enemies to weaken them. This would be a huge shift from his normal abilities and add a new moral dimension to his character as he decides when and how to use such a power.
With the Rinnegan, he could have abilities like the Six Paths techniques. He might be able to use the Preta Path to absorb chakra attacks. And with the Hyoton, he could create ice - based jutsu that are much more powerful than normal ice jutsu, like freezing an entire area instantly.
He could gain the ability to freeze Grimm. In RWBY, silver eyes have the power to harm and sometimes even stop Grimm in their tracks. Naruto's already impressive combat skills combined with this would make him a formidable force. He might also gain enhanced vision, like being able to see in the dark better or detect hidden threats more easily.
He could probably gain the ability to phase through objects. Just like wraiths are often depicted as being able to pass through solid things. This would give him an edge in combat and also in sneaking around.