He'd probably make new friends along the way. Naruto has a friendly and outgoing personality. He could meet other outcasts or people with similar experiences who would help him. These new friends could provide him with emotional support and practical help, like sharing their food or giving him a place to stay for a while.
Naruto might rely on his ninja skills that he's learned. He's a resourceful character, so he'd use his stealth and combat abilities to find food and shelter. For instance, he could hunt small animals in the forest and build a simple shelter using his knowledge of basic survival techniques.
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.
Naruto could meet other characters who have their own stories of hardship. By sharing their experiences, he realizes that he's not alone. This gives him the strength to face his emotional turmoil and move forward. He starts to build a new sense of self - worth based on his own actions and the relationships he forms outside of his family.
Running away from his family can make Naruto more independent. He has to rely on himself, which would strengthen his self - reliance. He'll learn to make decisions without the influence of his family, and this could make him a more decisive ninja.
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.
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.
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.
He might focus on his training more. By pouring all his energy into getting stronger, he can distract himself from the pain of family neglect. For example, he could spend long hours practicing new jutsu or improving his physical abilities.
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.
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.
The villagers who once shunned Naruto might start to realize his importance. They would see how much they took his presence for granted. His absence could make them reflect on their past behavior towards him. Some might even join the search parties out of guilt or new - found respect.