If Naruto isn't Minato's son in the fanfic, it could also mean that the Nine - Tailed Fox was sealed into him for a completely different reason. Maybe he was just in the wrong place at the wrong time, or there could be a conspiracy behind it. This would open up a whole new set of plotlines. Naruto would have to deal with the stigma of being the host of the Nine - Tailed Fox in a different context, without the family connection to Minato to either help or hinder him.
In this fanfic, if Naruto isn't Minato's son, the story might focus on Naruto's search for his true identity. He could be a part of some secret experiment or a lost clan. This would lead to him exploring different parts of the ninja world to find out where he really comes from. His relationships with other characters would also change. For example, his bond with Iruka sensei might become even stronger as Iruka could be one of the few people who truly believe in him without any connection to his supposed 'parentage'.
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.
He could be the son of a rogue ninja who abandoned him in Konoha for some unknown reason. This would add an element of mystery to his past.
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.
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.
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.
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.
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 could also show Naruto being taken in by other characters, like Iruka at a very early age. Iruka, seeing Naruto's situation, decides to be like a father figure to him. Naruto might then develop a different set of values based on Iruka's teachings rather than the absent influence of Minato and Kushina. He could become more focused on family values within the village, seeing the village as his true family since his biological family left him.
Minato could be developed further by exploring his time as the Fourth Hokage more deeply. For example, show his political decisions and how they were influenced by the presence of the Kyuubi. Also, his relationship with other villages could be explored, especially in relation to the Kyuubi's power. Maybe he had to make some tough calls to protect the village from those who might want to use the Kyuubi for evil.
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, Naruto has had the advantage of learning from many great shinobi over time. He also has a unique bond with his friends which could potentially give him more strength. Minato, on the other hand, had a shorter lifespan to develop his powers. Naruto has had more time to grow and surpass his father in strength.