He might meet the spirit of an ancient ninja who was the keeper of the library. This spirit could guide Naruto through the library and teach him things that no one else in the present day knows.
Perhaps in this fanfiction, the secret library is guarded by some sort of magical or powerful entity. Naruto, being Naruto, would try to befriend it first. Once inside, he could find prophecies about the future of the ninja world written in the books. This knowledge might put him on a new path of preventing future disasters and bringing about a new era of peace in the ninja world.
I recall a fanfic called 'Sentient Library: Naruto's New Ally'. In it, the sentient library is a sort of magical entity that has been observing the ninja world for ages. When Naruto finds it, it offers him knowledge that was lost over time. It shows Naruto different historical events through magical books, which gives him an edge in battles and also in understanding the politics of the ninja villages.
It could have Naruto discovering long - lost jutsu in the Uzumaki library. Maybe some ancient scrolls that hold the secrets of powerful chakra control techniques.
He could meet the previous Hokages. Since the library is infinite, there might be records or even some kind of spiritual manifestation of them. They could guide Naruto further in his journey.
Naruto could find a book that tells the story of a great hero who was just like him, full of energy and a desire to protect. This could inspire him to new heights. He might also accidentally trigger a trap or a hidden mechanism in the library while exploring, leading to an adventure filled with challenges and puzzles to solve.
Well, Naruto might discover some long - lost jutsu in that ancient library. Maybe there are scrolls that hold the secrets to new and powerful ninja techniques that no one has used in ages.
Perhaps Naruto would use the infinite library to understand his enemies better. He could study the weaknesses of his foes by finding information about their clans or fighting styles. Also, he might find books that could help him train his friends more effectively. The library could be a source of great power for Naruto, not just in terms of combat skills but also in terms of wisdom and leadership.
Well, he could gain new powers. Maybe the sentient library has books that teach him long - lost jutsu or give him knowledge about the nature of chakra that no one else in the ninja world knows. For example, he might learn how to combine different elemental chakra in a completely new way.
He could uncover secrets about his family. There might be some hidden records in the library that reveal things about his father or mother that he never knew before.