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.
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.
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.
In this fanfic, Sakura finally realizes her true feelings for Naruto. One sunny day, she nervously approaches Naruto and asks him out on a date. Naruto, surprised but happy, accepts. They go to their favorite ramen place first. As they sit and eat, they share stories and laugh. Then they take a walk in the park, enjoying the beauty of nature around them. It's a simple yet sweet date that marks the beginning of a new chapter in their relationship.
Perhaps in this fanfiction, Naruto meets Hiashi during a time of crisis in the Hidden Leaf Village. Hiashi, being an elder and a respected figure, is involved in the village's defense plans. Naruto, eager to help, wants to be part of the action. Hiashi initially doubts Naruto's abilities but is soon proven wrong. They then work side by side, with Naruto's Uzumaki chakra and Hiashi's Hyuga skills combining to create a powerful defense against the enemy.
The 'Hyuga's New Hope' fanfic is quite popular. It focuses on how Hiashi sees Naruto as a new hope for the shinobi world and encourages Hinata to be closer to him. It has a great mix of action and romance elements.
Maybe Hiashi adopts Naruto because he realizes that Naruto is a key to uniting the village. As Naruto becomes part of the Hyuga family, he starts to influence the clan's view on outsiders. He might introduce some new training methods that he learned from Jiraiya. Naruto could also have a role in protecting the Hyuga's secrets, like the Byakugan. He would grow in a different environment, and this would change his relationships with characters like Sasuke and Sakura in new and unexpected ways.
One interesting plot could be that Hiashi discovers Naruto has some hidden connection to the Hyuga clan. This would make Hiashi take a special interest in Naruto, and he might start training Naruto in Hyuga - related skills.
In some fanfictions, Naruto might use his newfound jutsu skills. He could quickly dash in between Hiashi and Hinata, using his speed and agility to block any attacks Hiashi might be launching towards Hinata. His Rasengan in his hand, ready to defend if Hiashi doesn't back down. This shows Naruto's determination to protect Hinata from any harm, even from her own father in this fictional scenario.