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.
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.
In this fanfiction, the plot could be that Naruto's kid is kidnapped by an old enemy of Naruto's who wants to train the kid to be against the village. The kid has to use his wits and the little ninja training he has received so far to escape. Meanwhile, Naruto is on a desperate search for his child. This would create a lot of tension and also show how much Naruto has grown as a father compared to his own orphaned past.
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.
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.
Training sessions could be a major factor in character development. Just like Naruto trained hard, his kid would too. During training, they could face difficulties and setbacks, which would teach them perseverance. They might also learn new jutsu and in the process, develop their own unique style, which would also shape their character.
Perhaps he feels pressured in Konoha. Having a kid adds more responsibility, and he might think that leaving Konoha will give him the space he needs to grow as a father and a ninja without the constant eyes of the village on him. He could be seeking a place where he can train his kid in a different way, away from the traditional Konoha methods.
On a more positive note, it could inspire some young ninja to step up. They would see Naruto's departure as a challenge to prove themselves and fill the gap he left. His leaving could also make Konoha re - evaluate its relationship with its ninja and how to keep them satisfied and in the village.
She would no doubt instill strong moral values in her child. Given her own experiences fighting against evil, she'd teach the kid to always stand up for what's right and to be kind to all magical beings. She might also teach the kid self - defense spells early on to protect themselves. For example, the Shield Charm and the Stunning Spell. And she'd make sure the kid understood when it was appropriate to use them. Also, Hermione would likely involve the kid in her research projects, sharing her passion for magical discovery with the next generation.
Well, if Naruto got the Rinnegan as a kid in fanfiction, it would completely rewrite the story. He'd have an edge in combat right from the start. With the Rinnegan, he could see chakra in more detail, which would make his learning of new jutsu easier. His opponents would be caught off - guard by his sudden access to such a powerful dojutsu. This could lead to some interesting plot twists, like him being able to solve problems in the village or against other ninja villages more quickly and efficiently.