There's also the possibility that it's an OC (Original Character) who is from a different village but travels to Konoha just to be trained by Kakashi. This OC could bring in new fighting styles and cultural elements to the story.
It might be one of the existing minor characters from the Naruto series who finally gets a chance to shine as Kakashi's apprentice. For example, that one ninja from the background who always admired Kakashi from afar and now has the opportunity to be his student.
It could be an original character created specifically for the fanfic. Maybe a young ninja with a unique kekkei genkai that Kakashi sees potential in.
Kakashi himself would also be a central figure. His role as a mentor would be crucial. He could be shown guiding the apprentice, sharing his wisdom and his own experiences as a ninja, like his time in the ANBU or his relationship with his former teammates.
The apprentice could be a new genin who shows great potential in the field of intelligence gathering. Maybe someone with a sharp mind and an ability to stay calm under pressure, like Shikamaru. He has the intelligence and strategic thinking that would make him a good candidate for Ibiki's apprentice.
It could be a rogue ninja from a rival village who is after the apprentice's special abilities. This rogue ninja might see the apprentice as a threat or a tool to achieve their own evil goals.
In this fanfiction, the plot might revolve around the relationship between Naruto and Kakashi as master and apprentice. Kakashi could be guiding Naruto not just in combat skills but also in understanding the deeper meaning of being a shinobi. They might go on special missions together where Naruto has to rely on Kakashi's wisdom and training more than his own brute strength. Also, Kakashi might share some of his own past experiences with Naruto, which would help Naruto grow both as a fighter and as a person.
Perhaps a powerful shinobi from another village who has a special grudge against the Leaf's ANBU. He targets Kakashi and Naruto specifically because of their reputation within the ANBU. He has unique abilities that pose a great challenge to them, like the ability to nullify ninjutsu for short periods.
Perhaps a storyline where the apprentice gets involved in a mission that was originally meant for Kakashi. This could lead to the apprentice having to prove themselves not only to their teammates but also to Kakashi. They might encounter powerful enemies and have to use their wits and the skills they've learned from Kakashi to survive. In the process, they could also discover some secrets about Kakashi's past that helps them better understand their master.
It could be a powerful rogue ninja who has a vendetta against Kakashi and sees his daughter as a way to get revenge. This antagonist might try to use her as a bait to lure Kakashi or Naruto into a trap.
Another option could be Luna Lovegood. She has a very unique perspective on magic already. If she were Merlin's apprentice, she could bring her eccentric ideas and combine them with Merlin's ancient magic. Her friendship with Harry Potter and the others could also play a big role in how she uses her newfound status as Merlin's apprentice to help the wizarding world.
Maybe an orphan from a non - Uchiha clan who has a special ability that Madara finds interesting. This character would be malleable and eager to learn from Madara, as they have no other real family or guidance. They could bring a fresh perspective to Madara's teachings.