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.
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.
The Sage of Six Paths himself could be the key character. Since the Rinnegan was originally his power, he could appear in the fanfic in some form, like a spirit or a time - traveling version of himself. He could then directly bestow the Rinnegan upon Naruto, seeing Naruto's potential and the need for such power to maintain the balance in the ninja world.
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.
His character would have a darker edge. The vampire nature could bring out a more primal and possessive side. And the Rinnegan would give him new levels of power and knowledge. He might start to question his old values as he realizes the potential of his new powers. He could become more solitary, training alone to master his combined abilities rather than relying on his friends like he used to.
He could become more confident, bordering on overconfident at times. With the Rinnegan, he has access to incredible powers, and this newfound strength might make him believe that he can handle any situation alone. However, his friends would surely keep him in check. For example, he might initially rush into a battle thinking his Rinnegan - enhanced abilities will ensure victory without fully considering the risks, but then realize his mistake when his friends point it out.