One reason could be to explore the consequences of a broken family dynamic in the Naruto universe. Minato was a hero but if he had neglected Naruto, it would have added a layer of complexity to Naruto's already difficult life. It also allows for character development as Naruto has to overcome this additional hurdle on his journey to becoming a great ninja.
Minato's presence can have a huge impact. He can act as a mentor, teaching Naruto things he never knew about his family and heritage. His wisdom and skills can inspire Naruto to reach new heights in his training and ninja development.
Well, Minato might constantly be by Naruto's side in the fanfiction. He could prevent Naruto from taking on any dangerous missions, even if Naruto is capable. For example, he may interfere when Naruto tries to join a risky battle against a powerful enemy, saying it's too dangerous for him.
One reason could be guilt. Since Minato wasn't there for Naruto when he was growing up in the original story, in the fanfiction, he might be overcompensating for that lost time. He feels like he has to make up for all those years he wasn't around to protect his son.
In a fanfiction of Hiruzen's daughter and Minato, perhaps Hiruzen's daughter is initially in awe of Minato's skills. Minato, on the other hand, is kind and respectful towards her. Their relationship could develop when they are paired up for a special mission. During the mission, they learn to trust each other completely and start to have feelings for one another. This could lead to a beautiful love story set in the world of ninjas.
He could feel it through a shared chakra link. Since they are related and Minato sealed the Nine - Tails in Naruto, there might be a latent connection that gets activated in the fanfiction, allowing him to sense Naruto's pain on a physical and emotional level.
Emotionally, Naruto would become more confident and stable. Having his father back would fill a void in his heart. This new - found emotional strength could translate into better chakra control and more powerful jutsu. Minato could also guide Naruto in handling the power of the Nine - Tails more effectively.
Minato would probably be very proud. He might notice Naruto's use of his own jutsu and be impressed by how Naruto has developed his skills. Kushina would be more emotional, maybe crying at the sight of Naruto growing up so well.
One touching moment could be when Minato's spirit appears right in front of Naruto when he is on the verge of being defeated by a powerful enemy. Just seeing his father's figure gives Naruto new strength and hope, which shows Minato's protectiveness in a very emotional way.
His speed would likely be a huge advantage. He could be faster than most wizards in combat or when running from danger. For instance, in a duel, his quick reflexes would allow him to dodge spells more easily.