The relationship can be developed by having Minato appear in Naruto's memories more vividly. For example, Naruto could start having flashbacks of Minato's training sessions or family moments that he didn't know about before. This would make Naruto feel closer to his father.
Well, the story could develop in many ways. Maybe Naruto could be an orphan found by someone else in the village, like an old ninja who decides to train him in secret. This would change Naruto's backstory completely. He might not have the same kind of pressure or expectations that come with being the son of a famous Hokage. His character could grow in a different direction, perhaps being more independent from the start as he has to create his own identity without the shadow of his 'father' Minato.
One idea could be Minato being able to communicate with Naruto in his dreams before Naruto discovers his true identity. This could offer Naruto some guidance and comfort during his difficult times in the village. Another plot could involve Minato training Naruto in a special ninja technique that only he knew, passed down from father to son. It could be a powerful jutsu that Naruto then uses to protect the village in a crucial moment.
Yes, there are likely some fanfictions out there with such a concept. You can try searching on popular fanfiction websites like Fanfiction.net or Archive of Our Own. Just type in 'Naruto Father of Minato' in the search bar and see what comes up.
Well, the plot could involve Naruto being trained in a very different way. Since he's the son of Minato and Mikoto, he might be taught a blend of the Namikaze and Uchiha techniques. For example, he could learn to combine Minato's Flying Thunder God Technique with some basic Uchiha fire - style jutsu. And there could be a story arc about how he unites the two clans that his parents come from, which have had some historical tensions. He would have to use his unique heritage to bring peace between them.
One way could be that Naruto is an orphan found by someone else entirely. Maybe he's taken in by a group of traveling merchants who raise him. His skills and abilities could be developed through different means, like learning from the various people he meets during his travels with the merchants.
The story would change a great deal. For one, Naruto wouldn't have the same connection to the Nine - Tails as he does in the original. Minato sealed the Nine - Tails into Naruto, so without that relation, the whole dynamic of the Nine - Tails within Naruto would be different. He might not have the same power - up moments related to the Nine - Tails.
Well, in such a fanfic, Naruto could be the son of someone like Jiraiya. Jiraiya was very close to Naruto and taught him a great deal. He might have had a secret relationship that led to Naruto's birth. Another possibility could be that Naruto is the son of an unknown shinobi from a far - off village who somehow ended up in Konoha.
Naruto would likely be more confident from the start. Since he has the combined blood of Minato and Mikoto, he might not have the same insecurities as in the original story. He would probably be more aware of his potential and be more self - assured in his abilities.