It might be that his family is overprotective. Naruto is a very independent character. If his family in the fanfiction is constantly trying to hold him back from taking risks or going on adventures, he may feel suffocated. Running away would be his way of breaking free and exploring the world on his own terms.
One possible reason could be that he feels misunderstood. In the Naruto series, Naruto often faced isolation and judgment from the villagers. If this extended to his family in the fanfiction, he might run away to find a place where he is accepted. Maybe he discovers some dark secrets about his family that make him feel unsafe or betrayed.
He could be running away because of false accusations within the family. Just like how he was wrongly judged by the village in the original story, if his family wrongly blames him for something he didn't do, like stealing a family heirloom or causing harm to a family member accidentally, he might choose to run away rather than face their unjust wrath.
One possible reason could be the constant feeling of being an outcast in the village. Despite his efforts, many villagers still look at him with suspicion because of the Nine - Tails inside him. He might feel that running away would give him a chance to find a place where he is truly accepted.
Another reason could be for the sake of protecting the village. Maybe he discovers some sort of threat that he believes he can only face by leaving Konoha. He doesn't want the village to be in danger because of him, so he runs away to draw the danger away from Konoha.
One possible reason could be that he feels misunderstood. In the village, he has always faced discrimination because of the Nine - Tails inside him. People often look at him with fear and suspicion. So, he might run away to find a place where he can be accepted for who he is.
One possible reason could be that he feels misunderstood by his family. Maybe they have different expectations of him and he can't bear the pressure, so he decides to run away to Kumo to start a new life.
He might meet an old hermit who teaches him a unique and powerful jutsu that his family never knew. This new jutsu gives him an edge and also makes him more confident in his abilities. He could then use this newfound power to help those in need, all while staying away from his family.
It could be due to a family secret. Suppose his family was involved in something dangerous or unethical that Naruto couldn't bear. For example, if they had ties to an enemy organization that Naruto discovered. He is a character with a strong sense of justice, so running away would be a way to distance himself from their actions and figure out how to deal with the situation on his own.
One possible reason could be feeling ostracized. In the Naruto world, being different is often not easily accepted. As a female Naruto, she might face even more scrutiny and prejudice. Maybe she's tired of the constant expectations and the way she's treated in Konoha, so she decides to run away to find a place where she can be herself.
It could be that Naruto discovers some secrets within his family that make him feel uncomfortable or out of place. For example, if there are hidden agendas or dark pasts that he doesn't want to be associated with, he might choose to move away to start anew.