At first, the apprentice might see Naruto as an enemy or at least a nuisance. However, Naruto's determination and his ability to never give up could intrigue the apprentice. As they are forced to work together on various tasks, they would start to understand each other better. Naruto could learn about the apprentice's loyalty to Zabuza and the apprentice could learn about Naruto's loyalty to his friends and village. This mutual understanding would lead to a deeper relationship.
One possible plot could be that Naruto meets Zabuza's apprentice during a mission. The apprentice, having grown up with Zabuza's teachings, initially has a rather cold and ruthless attitude. Naruto, with his usual optimistic and friendly nature, tries to befriend the apprentice. This could lead to the apprentice slowly changing their views on life and friendship, and perhaps they end up joining Naruto on his journey to protect the village.
A minor character like a blacksmith's apprentice could find Zabuza's sword. Instead of using it for fighting, they study it closely. They discover a unique metal alloy in the sword that they use to create new and improved weapons for their village. This not only changes their own fate as they become a renowned blacksmith, but also the fate of the village as they now have better weapons to defend themselves.
Well, another place could be Naruto - related forums. There are often threads where fans share and discuss their favorite fanfics. Some might be specifically about Zabuza's sword. You can join in the conversation and ask for recommendations too.
Naruto could develop his strategic thinking skills. By scaring Zabuza, he realizes that he can use psychological warfare in addition to his regular combat skills. So he starts to study his opponents more closely to find their weaknesses.
His personality would likely be a mix of Zabuza's stoicism and his own inherent determination. He might not show his emotions as freely, but deep down still have that strong will to prove himself. He could also be more distrustful of others at first, having grown up with Zabuza who is a bit of a loner. However, as he gets older, he might start to break out of that shell and show more of his true, kind self.
Another plot could be that Zabuza's sword ends up in the hands of an enemy of Konoha. Konoha sends a team of ninjas, including some of the younger generation like Naruto and his friends, to retrieve it. Along the way, they learn about Zabuza's legacy and the true strength of the sword. They have to battle the enemy who has already started to unlock some of the sword's powers, but in the end, they manage to get it back and decide what to do with such a dangerous weapon.
Well, perhaps Naruto takes Zabuza's sword and it becomes a symbol of his growth. He may face challenges from those who think he's unworthy of possessing such a weapon. Also, he could modify the sword to better suit his fighting style, like adding some of his own chakra -infused elements to it.