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.
One idea could be a time - travel fanfic where Sarutobi goes back in time and tries to prevent the rift between Naruto and Sasuke from ever happening. Another could be a fanfic where Naruto and Sasuke are transported to a parallel universe where Sarutobi is still alive and they have to work with him to get back home. Or a fanfic where Sarutobi trains Naruto and Sasuke together in a secret art that no one else in the village knows about.
One of the popular characters paired with Sarutobi is Jiraiya. They are both important figures in the Hidden Leaf and their relationship, whether it's as comrades or as teacher - student in some fanfic interpretations, is always interesting to explore.
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.
Perhaps he uses some ninja techniques like a shadow clone jutsu. He could make multiple clones to help him sort through and complete the paperwork quickly.
One characteristic could be the exploration of relationships. Since it's a fanfic, it might delve into relationships that Sarutobi has with other characters in the original work or even create new relationships. The 'lemon' aspect, if not referring to something inappropriate, could imply a certain zesty or interesting element in the story, like a plot twist or a unique interaction that adds flavor to the overall narrative.
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.