The maid could be an ordinary villager who Sarutobi thinks is trustworthy. Maybe she is a widow who needs a job and also has a kind heart, so Sarutobi chooses her to take care of Naruto.
One possible plot development could be Naruto being initially confused about having a maid. He might be a bit shy or not used to someone taking care of him in that way. The maid could also have a mysterious past that gets intertwined with Naruto's own journey. Maybe she has some hidden skills or knowledge that helps Naruto in his ninja training.
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.