Well, he could meet new friends or allies who help him grow stronger. For instance, he might encounter a group of outcasts who understand his situation and teach him new skills or ways of thinking.
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.
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 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.
Sakura would be very worried. She has always cared about Naruto, even if she sometimes gets annoyed with his antics. She would probably try to find him right away, thinking that he's in some kind of trouble.
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.
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.
He might miss out on some crucial training. Jiraiya has a lot of knowledge and skills to pass on, and by running away, Naruto could fall behind in his development as a ninja.
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.
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.