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.
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 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.
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.
The compilation of family trees usually did not include grandchildren in the family tree. According to tradition, a family tree was linked to the family name, and only the descendants of direct blood relatives would be recorded. Grandsons were members of families with different surnames. Although there was no difference in blood relations between grandchildren, they were considered to have different identities in the compilation of family trees. However, modern society was gradually open and compatible with the way genealogy was written. Some families would include their daughters, sons-in-law, grandchildren, and granddaughters in the genealogy to adapt to the development of the times and the changes in family structure. Therefore, it was up to the individual and the family to decide whether or not to write their grandson into the family tree. There were no fixed rules.