In fanfictions, Naruto could run away from his family due to emotional abuse. If his family members constantly put him down, criticize his abilities, or make him feel unworthy, he'd likely want to escape. Also, if his family is involved in some political intrigue that he doesn't want to be dragged into. For instance, they might be making deals with other clans that Naruto disagrees with. Additionally, if there is a new family member introduced, like a step - sibling who is favored over him and he feels unloved, that could prompt him to run away.
Maybe he feels misunderstood by his family. In many Naruto stories, his family situation can be complex. If he's 'godlike', he might have powers that his family doesn't understand or fears, leading him to run away to avoid conflict or being treated differently.
He could become more withdrawn. Since he's running away from family, he may lose some of his usual outgoing nature. For example, he might stop being as eager to make friends or be part of a team. He may also become more distrustful. After leaving his family, he might start to question the motives of others more, suspecting that people are like his family in some way.
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.
Perhaps Naruto feels that Kushina is being overprotective. He wants to prove his independence and go on his own adventures without her constant supervision, so he decides to run away.
One possible reason could be that Naruto might be afraid of the intense training Jiraiya has in store for him. Jiraiya is known for his strict and grueling training methods, and Naruto, being young and sometimes impatient, might just want to escape that for a while.
One possible plot could be that Naruto feels constantly overshadowed by his family's expectations. Maybe his family is full of powerful shinobi and they expect him to be just as great immediately. He runs away to find his own path and prove himself on his own terms. Along the way, he meets new friends who are also outcasts and together they form a sort of 'family' that accepts him for who he is.
Maybe he's running away from the pressure in the village. There are always high expectations on him, and he might feel suffocated and need some time alone to figure things out.
The family would be shocked at first. They might start to question their actions towards Naruto. If they were too strict or didn't give him enough love, they would feel guilty. For example, his father may realize that he pushed Naruto too hard to follow a certain path and now regrets it.
One possible reason could be fear. Minato is a very powerful shinobi, and Naruto might be scared of him for some reason. Maybe he has heard stories about Minato that made him afraid.