Well, in many such fanfictions, Naruto might start learning his father's jutsu when he discovers some hidden scrolls or when a mentor who knew his father decides to teach him. Maybe it could be someone like Jiraiya who reveals some secrets about his father's techniques.
In some fanfictions, Naruto starts learning his father's jutsu after he unlocks a new level of his chakra. This new power allows him to access memories or knowledge related to his father's techniques. It could also be that an enemy with a similar jutsu attacks the village, and Naruto realizes that learning his father's jutsu is the key to defeating this foe. So, he embarks on the journey of learning the jutsu with the help of his friends or some old ninja texts.
One challenge could be the complexity of the jutsu itself. His father's jutsu might be very advanced and require a high level of chakra control that Naruto doesn't initially have.
Maybe he found some hidden scrolls left by his mother. These scrolls could have contained the details of her jutsu. Naruto, being the curious and determined ninja he is, would have studied these scrolls carefully and learned the jutsu.
He might learn a time - traveling jutsu. This would be really cool as it could allow him to go back and change certain events in the Naruto world, or gain knowledge from the past to use in the present battles.
He could potentially learn the Flying Thunder God Technique. It's a very powerful jutsu that Minato was famous for. Naruto's already fast with his Sage Mode and Nine - Tails Chakra, and this would make him even more formidable in battle.
In many fanfictions, it could be through finding old documents or letters. Maybe he stumbles upon a hidden scroll in his home that contains some information about his father. For example, it might be a letter from his father to his mother that gives some clues about his father's personality or his role in the village before Naruto was born.
If Naruto creates a very powerful and dangerous jutsu, some characters might be afraid of him. For example, the villagers who were once suspicious of his Nine - Tails power could again start to fear him. However, his close friends like Sasuke and Sakura would likely trust him and help him control and master the new jutsu.
Well, it could be the Uzumaki Chain jutsu. In the Naruto lore, the Uzumaki clan had some unique abilities and the chain jutsu was one of them. His mother Kushina was from the Uzumaki clan, so it's possible that's the jutsu he learned in the fanfiction.
Well, in certain fanfic scenarios, Naruto could learn paper jutsu as a result of some experimental training. For example, he might be trying to combine different elements of his existing jutsu and accidentally discovers a way to manipulate paper. It could be that his chakra takes on a new form during a particularly intense training session and he realizes he can control paper with it.
In fanfiction, Naruto learning the Vacuum Jutsu could be really cool. It might give him a new and powerful offensive technique. Maybe he'd learn it from an unexpected source, like an old hermit in a forgotten land. This could potentially change the power dynamics in battles he's in.
If he learned a healing jutsu, he would become more nurturing. He could use this jutsu not only to heal his friends in battle but also to help the villagers. His reputation would change from just a powerful fighter to also a healer, and he would be more respected in a different way.