Naruto's character would change in multiple aspects. First, he would have to adapt to the new abilities of the Rinnegan. This would require him to study and train more, making him more focused and perhaps a bit less impulsive. Second, the power of the Rinnegan might give him a different perspective on the ninja world. He might start to see the power struggles and politics in a new light. Third, his relationships with his friends, especially Sasuke, would be different. Since he now has a part of Sasuke (in the form of the Rinnegan), their bond could become even more complex and deep. And finally, he might face new challenges in terms of his identity, as he is now not just the Nine - Tails Jinchuriki Naruto, but also the wielder of the Rinnegan.
In some fanfiction concepts, it might be due to a unique chakra interaction. Maybe the Kyuubi's powerful and ancient chakra somehow triggers a latent ability within Naruto that results in the manifestation of the Rinnegan. It could also be part of some kind of prophecy or destiny within the fan - created world of the story.
It could also be a situation where Naruto's connection to Hagoromo is deepened in the fanfiction. For instance, Naruto might have a vision or a spiritual encounter with Hagoromo. During this encounter, Hagoromo realizes that Naruto is the key to bringing true peace to the ninja world. So, he imparts the power of the Rinnegan to Naruto to help him achieve this grand goal. This would be a more complex and involved process, perhaps involving Naruto learning about Hagoromo's past and his own role in the grand scheme of things.
In some fanfiction concepts, there could be a spiritual or mental connection established between Naruto and Nagato. Since Naruto has this unique ability to understand and connect with people on a deep level. Through this connection, Nagato might pass on the power of the Rinnegan to Naruto. It could be like a transfer of not just the physical power but also the knowledge and understanding that comes with the Rinnegan.
In some fanfictions, it could be through a strange and powerful jutsu transfer. Maybe Madara, on the verge of defeat, decides to transfer his Rinnegan power to Naruto for some unforeseen reason, like seeing a new path for the ninja world in Naruto.
Naruto's perspective on the ninja world could change. With the Rinnegan, he would see things in a different light, both literally and metaphorically. He might have to deal with new enemies who are attracted to his new power or new responsibilities that come with such a powerful dojutsu. His relationships with his friends and allies could also be affected as they may view him differently now that he has this extremely powerful ability.
He might become much more powerful. With the Rinnegan, he could gain new abilities like the Six Paths powers, which would make him a force to be reckoned with in battles.
In some fanfiction concepts, the chakra of the Kyuubi is not just pure fox chakra but also contains remnants of the Sage of Six Paths' power. Since the Rinnegan is related to the Sage of Six Paths, when Naruto reaches a certain level of power and harmony with the Kyuubi, the dormant Rinnegan - like power within the Kyuubi's chakra awakens in Naruto. This could be a result of Naruto's strong will and his journey to master the Kyuubi's power which eventually leads to him obtaining the Rinnegan.
One interesting plot could be Naruto using the Rinnegan's powers in unique ways during battles. For example, he might combine the Rasengan with the Deva Path's ability to attract and repel, creating a new and powerful attack.
In this fanfiction, Naruto might gain incredible new powers. He could have enhanced control over all the elements and maybe even the ability to manipulate life and death like the true Rinnegan is often associated with in the Naruto lore. This would likely make him an even more formidable opponent and a great protector of the Hidden Leaf Village.