Maybe in the fanfic, there is a dark force that tries to manipulate Hiashi into attacking Naruto. The fanfic could explore how Naruto tries to reach out to Hiashi, to show him that they are being played. Along the way, they have an intense battle where Naruto shows off his new jutsus and Hiashi uses his clan's most secret techniques. In the end, they realize they were on the same side all along.
Well, Naruto has his powerful jutsus like the Rasengan and his ability to use the Nine - Tails' chakra. However, Hiashi is a very strong Hyuga with the Byakugan. In a fight, Naruto's unpredictable and powerful attacks could give him an edge. But Hiashi's precision and knowledge of Gentle Fist techniques might also make it a tough battle. So, it's really hard to say who would win definitely.
Hiashi could be initially skeptical. Naruto has a reputation for being a bit of a troublemaker at times. But if Naruto presents a good case, Hiashi might warm up to the idea. For instance, if Naruto shows how his actions would bring honor and prosperity to the Hyuga clan.
In this fanfic, Naruto might save Hiashi in a battle situation. Maybe Hiashi is facing a powerful enemy that he can't handle alone, and Naruto steps in with his strong ninja skills. For example, if Hiashi is being attacked by a group of rogue ninjas, Naruto could use his Rasengan to fend them off.
One possible reason could be that Naruto is in love with a Hyuga clan member. Since Hiashi is the head of the Hyuga clan, Naruto might think it's proper to ask for his blessing. For example, if he has feelings for Hinata, Hiashi's daughter. Hiashi's approval would mean a lot in terms of family and clan acceptance.
Maybe Hiashi is poisoned, and Naruto has to find an antidote. He could go on a journey to find a rare herb or seek the help of a medical ninja. Along the way, he might face various obstacles, but his will to save Hiashi would keep him going. He could also use his connection with the Nine - Tails' chakra to boost his search or fight against any threats.
There are fanfics that focus on a political alliance between Naruto (as a future Hokage) and Hiashi. Hinata is caught in the middle of this in a way that she has to prove her worth not just as a Hyuga but also as someone who can stand beside Naruto. In these types of fanfics, there's often a lot of character development for all three as they navigate through the complex world of shinobi politics.
One possible motive could be a misunderstanding. Maybe Hiashi mistakes Naruto for an enemy who has infiltrated the village in disguise. Another reason could be related to clan politics. If Naruto somehow gets involved in a situation that threatens Hiashi's clan interests, Hiashi might take extreme measures.
Naruto might convince Hiashi by pledging his loyalty to the Hyuga clan. He could say that he'll uphold the clan's values and traditions. Also, Naruto could mention how Hinata has been a positive influence on him, which shows his respect for her. He might also talk about his plans for the future and how Hinata will be an important part of it in a way that benefits both the Uzumaki and Hyuga clans.
Well, it might be a fanfic that delves into a new adventure. Naruto, being the main character, could lead the group. Hinata would be by his side, with her shy but strong personality. Hanabi could represent the new generation of the Hyuga clan, learning from Hiashi. And the Uzumaki connection could add a layer of mystery and power to the story. For example, they might discover an ancient Uzumaki technique that helps them in a battle or a crisis.