It's possible that the kid is a result of some sort of time - travel or alternate reality situation. Maybe the kid is actually Naruto from a different timeline or dimension, and Naruto has to figure out how to deal with this strange situation while on his training trip.
Well, it could be that the kid serves as a source of motivation for Naruto during his training. He would want to become even stronger to protect his child. Also, the presence of the kid could lead to some comical situations, like when the kid tries to imitate Naruto's training moves in a clumsy way.
In the fanfiction's training trip, Naruto may encounter some spiritual beings. These beings could be related to the origin of chakra or some long - forgotten ninja legends. Interacting with them would be a very different experience for Naruto. He would have to use not only his physical skills but also his wits and understanding of the ninja world's deeper mysteries to communicate with them. This could lead to a major transformation in his ninja journey, as he delves deeper into the spiritual side of the ninja arts.
Perhaps the kid is from a foreign land that Naruto visited during one of his missions. The kid could be an outcast in his own place because of some unique trait. Naruto, being the kind - hearted person he is, decides to adopt him. It could be that the kid has a different kind of chakra or a strange mark on his body that makes others afraid of him, but Naruto doesn't care.
He could meet an old hermit ninja who imparts his wisdom and knowledge to Naruto. This hermit might have been living in seclusion for years, but seeing Naruto's potential, decides to teach him things not found in normal ninja training.
Naruto might meet some foreign ninjas from far - off lands that have completely different fighting styles. These ninjas could teach him new ways to use his chakra and improve his combat abilities. He may also encounter some mysterious hermits living in the mountains or forests who possess great wisdom and unique jutsu. They could guide him on his journey of self - improvement and give him valuable advice on how to become stronger as a ninja.
He could meet other sannin. Maybe he stumbles upon Orochimaru who is on some secret mission. This encounter would be full of danger and Naruto would have to use all his skills and wits to survive.
He could meet some old enemies who have changed their ways. For instance, some of Orochimaru's former minions might have decided to start anew and cross paths with Naruto. Or perhaps he meets a hermit ninja who has been living in isolation for years and imparts some valuable wisdom to him.
It could be a relative of one of Naruto's old foes. For example, a nephew of Orochimaru who has been taught by Orochimaru's remaining followers. He might be jealous of the peace Naruto has brought and wants to disrupt it by targeting his kid.
The kid could be a young demigod whose parents are unknown. Maybe he's been living on the streets, hiding from monsters, until Percy finds him.
It could be a bully in the village. This bully could constantly pick on Naruto because of his status as the Jinchuriki of the Nine - Tails. He might try to steal Naruto's food or prevent him from playing with other kids.