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.
He might become more self - reliant. Without the immediate support of his family, Naruto has to learn to fend for himself. This could lead to a toughening of his character and a more independent mindset.
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.
One reason could be that he feels misunderstood by the villagers. Despite all he has done for Konoha, some still hold grudges or are afraid of his power. Another might be that he wants to explore the world on his own terms, away from the responsibilities and restrictions of Konoha. Maybe he also feels a sense of loneliness there, as he has lost many important people and the village constantly reminds him of those losses.
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.
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.
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.
They would probably start searching for him. This could lead to the family members getting closer as they united in their goal to find Naruto. Or it could cause conflicts among them if they blamed each other for Naruto's running away.
His departure would definitely put a strain on his relationships. His close friends like Sasuke, if he is back in Konoha at that time, would have mixed feelings. They might worry about Naruto but also be a bit angry that he left without consulting them. Naruto's younger friends like Boruto would be confused and miss him a great deal, and this could lead to some distance in their relationships.
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.