One of the most popular is 'Naruto Shippuden: Sakura Hiden - Thoughts of Love, Riding Upon a Spring Breeze'. It's popular because it shows Sakura in a new light, with her own adventures and growth.
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.
I don't think it's strange at all. Naruto is a very curious person. Fanfiction can offer new perspectives on his own life and the lives of his friends. He might learn something new about himself or get inspired by the creative ideas in the fanfiction. It could also be a form of entertainment for him during his downtime.
A Naruto x Hinata fanfiction could focus on a mission they go on together during Naruto Shippuden. They are forced to rely on each other in a dangerous situation. Hinata gets injured protecting Naruto, and Naruto is filled with a new - found determination to protect her. This experience deepens their bond, and Naruto starts to see Hinata in a whole new light, leading to the start of their romantic relationship.
Another interesting novel is 'Naruto Shippuden: Kakashi Hiden - Lightning in the Icy Sky'. It focuses on Kakashi's story and his past, which is really fascinating. There are a lot of details about his relationships and his growth as a ninja.
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.
Well, Kurama's sad story in Naruto Shippuden is complex. He was feared by the villagers in the Hidden Leaf because of his power. When he was sealed into Naruto, he was angry and tried to take over Naruto's body multiple times. His past was filled with being hunted and then trapped. It wasn't until he started to understand Naruto's will and dreams that he began to change. Naruto's unwavering determination and kindness towards him slowly melted his cold heart.
In the end of Naruto Shippuden, there's a sense of closure as major conflicts are resolved. Naruto becomes a respected leader, and many characters find their places and happiness. The bonds among friends and the growth of characters are emphasized.