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 relationship would be very different. Instead of being an orphan, Naruto would have his parents around for all the important moments in his life. Minato could teach him about being a Hokage from an early age, and Kushina could give him the motherly love and advice. There would be a strong family bond. Naruto would be more confident in his abilities because he has his father, a great ninja, to guide him. And he would have a more stable emotional state with his mother's care.
Minato would likely be extremely shocked. Naruto is his son, and the idea of Naruto attacking him would be completely unexpected. He would probably try to talk to Naruto first, to find out what is wrong and why he is being attacked.
If Minato and Mei were Naruto's parents in a fanfiction, the story would take a whole new turn. In the original, Naruto was an outcast because of the Nine - Tailed Fox sealed within him. But with Mei as his mother, there could be different reactions from the village. Also, Naruto's training would be different. He might learn from Mei's combat style which is more aggressive and fluid compared to what he learned in the original. And the mystery surrounding his parents would be completely different. Instead of just the legacy of Minato, there would be the added mystery of Mei's influence and her village's secrets that could be passed on to Naruto.
Naruto could inherit Minato's calm and intelligent nature. Minato was known for his quick thinking in battle. From Mei, he might get her boldness. Mei was never afraid to speak her mind and take risks. So Naruto could be a combination of being smart in combat situations like Minato and fearless like Mei.
One of the main similarities is their self - sacrifice. Minato sacrificed himself to save the village and his son. Naruto has also shown that he is willing to put himself in harm's way for the greater good. He's always the first to jump into a dangerous situation if it means protecting those he loves. Another aspect is their leadership qualities. Minato was a great leader as Hokage, and Naruto has developed into a great leader as well, inspiring those around him.
The emotional bond would be stronger. Naruto wouldn't have to face the pain of growing up without a father. Minato could be there to cheer him on during his training and be a source of comfort when Naruto faced difficulties, like when he was bullied as a child. He could also help Naruto better control the Nine - Tails' power from the start.
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.