I'm not sure specifically as there could be many novels with 'Minato' in the title. It could be about a character named Minato, or it could be set in a place called Minato. Without more context, it's hard to say.
I don't know the specific name of the Minato manga off the top of my head. It could be related to a particular genre or have some distinctive features that might help identify it.
If Minato survived, it might lead to a power balance shift in the ninja world. His presence could prevent some of the later conflicts or handle them differently. Also, his relationship with his family and friends would evolve in interesting directions.
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.
Minato could also teach Naruto about leadership. As the Fourth Hokage, he led Konoha. He could share his experiences on how to make tough decisions for the good of the village and its people. He might tell Naruto stories of past battles and how he strategized to win, which would help Naruto in his future as a leader in Konoha.
On a more personal level, they could face internal guilt. As Naruto grows up and achieves great things on his own despite their neglect, Minato and Kushina may realize what they've missed out on. This realization could cause them a great deal of internal turmoil and regret for not being there for their son when he needed them.
One common ability is their speed. Minato was known as the Yellow Flash for his incredible speed. Naruto, with his Sage Mode and later forms, can also move extremely fast. Another is their use of the Rasengan. Minato created the Rasengan, and Naruto has mastered it and even developed it further.
One possible reason could be for Naruto's protection. Maybe Minato knew of some great danger that was coming for Naruto and he thought that banishing him would keep him out of harm's way. Another reason might be related to some prophecy or hidden power within Naruto that Minato feared could be misused if Naruto stayed in the village.
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.