Naruto's character could develop a stronger sense of responsibility towards the village. Since his father was the Sandaime Hokage, he would feel a greater obligation to protect and uphold the values of Konoha. He might also study more about the village's history and politics, which would make him more well - rounded. He could become more diplomatic, as he would need to interact with different clans and ninja with different agendas. His sense of humor and determination would still be there, but they would be channeled in a more refined way.
He would probably be more confident. Knowing he is the son of the Sandaime, he has a strong lineage behind him. This could make him more assertive in his actions and decisions. For example, he might not be as hesitant to take on leadership roles in missions.
In this fanfiction, Naruto's character development could be deeply influenced by the legacy of his father. He would constantly be compared to the Sandaime, which could initially be a burden. However, as he matures, he would learn to use this as motivation. He might develop a more serious side, as he realizes the weight of his family name. At the same time, he could also become more empathetic towards the older generation who knew his father well. His relationships with his friends might also change, as they too would have to adjust to this new aspect of Naruto's identity.