He might gain better chakra control. Kakashi could teach Naruto how to precisely manage his chakra, allowing him to perform jutsus more efficiently and for longer periods. For example, Naruto could use his Shadow Clone Jutsu without exhausting his chakra so quickly.
He could potentially gain an enhanced understanding of wood - style techniques. Since Hashirama was a master of it, Naruto might learn how to create more complex and powerful wood constructs, like huge forests for both offense and defense.
He could gain better stealth skills. Hidan is quite good at sneaking around and Naruto might pick up on that. So he could move more quietly and be better at surprise attacks.
He might gain enhanced chakra control. Since Kami is a powerful entity, Naruto could learn to manipulate his chakra in ways he hadn't before, like creating more complex jutsus with less effort.
Well, it could be that in this fanfic, Kakashi discovers a new potential in Naruto. He might train Naruto in some ancient and forgotten ninja arts. This would set Naruto apart from his canon self. Also, their relationship could be explored more deeply, like Kakashi becoming a father - figure to Naruto, and this emotional bond could influence Naruto's growth as a ninja.
Naruto could gain the ability to use different forms of his Nine - Tails chakra more effectively. Since Killer Bee has great knowledge about Jinchuriki powers, he could teach Naruto how to access and use his chakra in ways he hadn't thought of before. Also, Naruto might learn new physical combat techniques from Bee, like improved hand - to - hand combat skills or new ways to use his kunai and shuriken.
He could gain the ability to sense death. Maybe he can tell when someone is close to dying or if there's a dangerous situation that could lead to death. It would be like a sixth sense.
Sasuke could be a side character. Since Naruto is getting special training from Kakashi, Sasuke might feel jealous or motivated to train harder himself. Sakura could also be there, perhaps she tries to learn from Kakashi's training methods to help Naruto better. Another potential side character could be Iruka, who might check on Naruto's progress from time to time and offer some words of encouragement.
He might gain the ability to manipulate ice like Haku. This could be a very useful addition to his existing skill set, especially in battles against fire - type opponents. He could create ice clones instead of the regular shadow clones, which would be more durable in certain situations.
Well, Kakashi might introduce Naruto to more strategic ways of fighting. For example, he could teach Naruto how to better analyze his opponents before attacking. Naruto, who was previously more impulsive, may start to hold back and look for the right moment to strike, like Kakashi does with his Sharingan.
In a future Naruto and Kakashi fanfiction, Naruto could gain the ability to communicate with all living things. This would help him in understanding the needs of the village and the surrounding nature. Kakashi, on the other hand, might develop a new Sharingan ability that allows him to see the future in a very limited way, like seeing the next move of his opponent a few seconds in advance. This would enhance his combat skills significantly.