In a fanfiction where Naruto has the Rinnegan, it might change his relationships too. His friends could be both in awe and a bit scared of his new power. The villages might see him as an even greater threat or a more reliable protector. Also, his training would have to be different. He'd need to learn how to control and master the Rinnegan's abilities, which are very different from the chakra nature he was used to.
With the Rinnegan, Naruto could also gain the power of the Human Path. This might allow him to read the minds of his opponents or extract information from them in a very unique way. He could also use the Naraka Path to torture his enemies psychologically, which is a very dark but powerful ability that the Rinnegan offers.
Another plot could be that the Rinnegan was secretly implanted in Naruto by an enemy as part of a long - term plan to turn him evil. Naruto then has to struggle not only with the new power but also with the dark influence it tries to exert on him. His friends initially suspect him, but he has to prove that he's still the same Naruto while also trying to figure out how to get rid of the Rinnegan's negative effects.
One implication is that he becomes much more powerful. The Rinnegan gives him access to powerful abilities like the Six Paths techniques, which would change the power dynamics in battles in the fanfiction. He could now face stronger enemies more easily.
With Kakashi, Naruto having the Rinnegan could change their teacher - student dynamic. Kakashi would be both amazed and a bit worried. He would need to guide Naruto in a new way, as the Rinnegan brings a whole new set of powers and potential dangers. Naruto might start to teach Kakashi some things he discovers about the Rinnegan, reversing their roles to some extent.
Another possibility is that it's a result of some strange chakra experiment gone wrong (or right). Scientists in the Naruto universe are always dabbling in chakra research. Maybe an experiment meant to enhance her natural abilities instead gives her the Rinnegan, and she then has to learn to control it quickly to avoid being consumed by its power.
If Naruto got the Rinnegan, he might become more cautious. The power of the Rinnegan is so great that he would realize he needs to be more careful with it. He wouldn't be as brash as before.
In Naruto fanfiction, the Rinnegan in Naruto is often portrayed in really cool and powerful ways. Some fanfics might give Naruto the ability to use all the Rinnegan powers to their fullest extent right away, like being able to effortlessly summon the Gedo Statue or use the Deva Path powers without any drawbacks. It's a way for writers to explore new storylines and make Naruto even more of a badass than he already was in the original series.
Well, at first it might cause some strain. His sister may feel overshadowed by Naruto's new power. But as time goes on, Naruto could use the Rinnegan to help his sister discover her own latent abilities. This would transform their relationship into one of mutual growth and support. He could use the powers of the Rinnegan to train her in unique ways, and she could in turn offer him emotional support, making their bond even deeper than before.
It might be related to a connection with some powerful entity in the Wave arc. For example, if there's a secret shinobi from the past whose power is tied to the Rinnegan and Naruto somehow forms a bond with that power source, it could lead to the awakening. Also, the intense emotions and challenges he faces in the Wave arc might be a catalyst, pushing his chakra to a new level and awakening the Rinnegan.