They're popular because it gives fans a chance to explore different possibilities for Naruto's growth. Instead of the same old canon story, an oc can introduce new ideas and training methods.
One popular 'oc raises naruto fanfic' might be 'A New Beginning'. In it, the original character has a unique backstory that influences how they raise Naruto. They might teach him different jutsu or ways of thinking that deviate from the norm in the Naruto universe. Another could be 'The Unlikely Mentor' where the oc uses their own set of skills, perhaps from a different land or clan, to shape Naruto into a stronger shinobi.
In this fanfiction scenario, Naruto raising Hanabi could bring about some interesting character developments. Naruto's positive and determined nature might inspire Hanabi to be more confident. He could teach her some of his ninja techniques, like the Rasengan in a modified form suitable for her. And perhaps he'd also share his values of friendship and never giving up, which would shape Hanabi into a more well - rounded and strong kunoichi.
There's 'The OC's Mysterious Power and Naruto'. Here, the OC has a mysterious power that is not fully understood even by the most powerful shinobi in the story. When they start raising Naruto, they begin to share bits of this power with him. Naruto then has to learn how to control and utilize this new power, which is very different from the normal chakra - based powers in the Naruto universe. This makes for an exciting and fresh take on Naruto's growth story.
An 'oc raises Harry Potter' fanfiction can be unique by giving the oc a really distinct personality. For example, if the oc is a grumpy old wizard who doesn't follow the norms of the wizarding world but still manages to raise Harry in his own way. Another way is to change the setting. Instead of raising Harry in England, the oc could take him to a different magical country and expose him to different magical cultures. Also, the oc's motives for raising Harry can be unique. Maybe they're not doing it out of kindness but for some self - serving reason that later changes as they get to know Harry.
Jiraiya could still play a role. Even if Naruto is raising himself, Jiraiya might cross paths with him and see his potential. He could offer some sporadic guidance without fully taking Naruto under his wing.
In such fanfictions, Naruto's character often develops a greater sense of responsibility. Raising someone or something forces him to think beyond himself and his own goals.
It could be a child of one of Naruto's friends who passed away. For example, if Shikamaru died in a battle, Naruto might take in his child. Naruto would then try to pass on Shikamaru's wisdom and strategic thinking to the child, along with his own ninja skills.
One possible plot could be that Naruto discovers an orphaned child during one of his missions. He decides to take the child under his wing and raise it as his own. He would teach the child ninjutsu, just like Jiraiya taught him, and also instill in the child the values of hard work and loyalty.
Well, Naruto would surely pass on his adventurous spirit to Hanabi. She'd probably be more willing to take risks and explore new areas of the ninja world. Naruto might also train Hanabi to be a better fighter. He could use his experience from his own training with Jiraiya and Kakashi to design a special training regime for her.