If Naruto were raised by Sarutobi in fanfiction, it could lead to some interesting character development. Sarutobi's calm and collected nature might rub off on Naruto, making him less impulsive. He could also introduce Naruto to different powerful ninja techniques earlier. Moreover, Naruto would likely have a stronger bond with the older generation of the village, as Sarutobi would encourage such relationships. This could change the way Naruto interacts with his peers and enemies alike.
Naruto's relationships would experience a significant shift. With Sarutobi as his guardian, he would have a closer relationship with the Anbu. He would be more involved in the inner workings of the village at an earlier stage. His relationship with Kakashi would also be different. Instead of Kakashi being his main mentor, Sarutobi would play that role. This would mean that Naruto would approach his training and missions with a different mindset, which in turn would affect how he relates to his comrades and opponents.
In this fanfiction scenario, Naruto might learn unique chakra control techniques from Sarutobi. Sarutobi, being an experienced ninja, could teach Naruto how to better manage his large chakra reserves. This would lead to Naruto being able to perform jutsus more efficiently and with less waste of chakra.
I really like 'Sarutobi's Unseen Trials'. In this fanfiction, it focuses on the political aspects that Sarutobi had to deal with in the Hidden Leaf Village. It gives a new perspective on his leadership and the difficult decisions he had to make.
Perhaps in this fanfiction, Sarutobi is trapped in a genjutsu created by a rogue ninja. Naruto, being highly resistant to genjutsu due to his Uzumaki blood, manages to break into the genjutsu world. There, he has to fight his own inner demons as well as the rogue ninja's illusions to free Sarutobi.
Obviously Naruto would be one of the main characters interacting with him. Naruto would look up to Sarutobi as a mentor and a source of knowledge. Their interactions could range from training sessions to deep conversations about the future of the village.
It could be that Naruto has a different view of leadership than Sarutobi. Naruto might see Sarutobi's leadership as being too passive or too focused on maintaining the status quo. He, with his more energetic and revolutionary ideas, may resent Sarutobi for not being more progressive in making the village a better place for everyone, including himself.
One idea could be that Naruto, as Sarutobi's apprentice, discovers a new jutsu hidden in the old Hokage's personal library. This jutsu could be a powerful fire - style technique that was once used by Sarutobi's master. Naruto would have to train hard to master it, facing challenges like controlling the intense heat. It would also add a new dimension to his combat abilities and make him stand out even more in battles against powerful foes.
I really like 'Naruto, Sasuke and Sarutobi: A New Beginning'. In this fanfic, it starts with Sarutobi having a premonition about the future of Naruto and Sasuke. So he takes steps to guide them differently. It's an interesting take on their relationship. 'Sasuke's Redemption under Sarutobi' is also good. It shows Sasuke trying to make amends with the village with Sarutobi's guiding spirit in mind. And 'The Hidden Power of the Trio' is a fanfic that gives a new perspective on the powers they possess and how they can be combined for the greater good.