One possible storyline could be that the Si OC is a shinobi from a village that has been wronged by Konoha in the past. They see Naruto as a symbol of Konoha's power and thus direct their hatred towards him. For example, if their village was raided by Konoha in the past for some reason, they hold a grudge against all things Konoha, including Naruto.
Perhaps the Si OC has a family member who was defeated or killed by Naruto in a battle. This would fuel their hatred towards him. The fanfic could then revolve around the OC's attempts to get revenge on Naruto. They might try to undermine him in various ways, like spreading false rumors about him in the ninja world or trying to sabotage his missions.
I'm not really sure of specific fanfics where the self - inserted original character (Si OC) hates Naruto. You could try searching on major fanfic platforms like FanFiction.net or Archive of Our Own. Look for tags related to 'original character' and 'hatred towards Naruto' to narrow down your search.
One possible storyline could be that Athena appears in the Naruto world and starts interfering with the ninja villages. She might try to impose her own set of laws based on Greek mythology, which Naruto sees as a threat to his people and their way of life, thus leading to his hatred. For instance, she could demand that the villages stop using chakra in a way she deems inappropriate according to her values.
Perhaps Naruto gets corrupted by some dark force. It changes his personality completely. He might start seeing all the people in the Hidden Leaf Village as the source of his problems. He could remember all the times he was ostracized as a child and now blames everyone for it. So in the fanfic, his hatred for everyone grows as he views the world in a new, darker light.
In the fanfic, Naruto could hate the Hokage because of some bad experiences related to past Hokages. Perhaps he discovers some dark secrets about how the Hokage office has made decisions that he doesn't agree with. It might be something like a cover - up of certain events that Naruto believes should have been dealt with more fairly.
One scenario could be that Naruto discovers some hidden secrets about Minato and Kushina that he interprets as them being selfish. For instance, he might find out that they had a choice to be with him but chose something else instead, which makes him angry and full of hatred towards them.
One reason could be creative exploration. Writers might want to see how the story would unfold if Naruto hated Hinata. It's a way to break away from the normal storyline.
Well, without actually reading it, it could be about an original character (OC) in the Naruto universe created by someone named Kamizuki. It might be a story that focuses on this OC's journey, their relationships with the existing Naruto characters, and how they fit into the Naruto world's storylines, like perhaps their adventures in different ninja villages or their part in the battles.
One possible storyline could be the OC sister starting at the same school as the main characters. She might initially struggle to fit in because she has a different personality compared to the tough - minded students around her. For example, she could be very shy and artistic, but through her relationship with the main characters, she gradually learns to be more assertive and stand up for herself.
One possible storyline could be Naruto being possessed by a Devilman - like entity. His friends then have to find a way to free him while dealing with the new, more dangerous Naruto. Another could be Naruto willingly accepting Devilman - like powers to defeat a greater threat, but then struggling to control them.
A possible storyline is that Naruto accidentally stumbles upon a hidden dojo where Rokushiki is practiced. The dojo master, seeing Naruto's potential, takes him in as a student. As he learns Rokushiki, he has to face the challenges of combining it with his ninja skills. He might also face opposition from some of the other students who are jealous of his natural talent. This could lead to various conflicts and growth for Naruto's character within the fanfic.