Well, it depends on the specific fanfic. Some fanfics might really emphasize Naruto's growth to be on par with Minato in terms of smarts. They could show Naruto learning from Minato's memories or having some sort of spiritual connection that transfers knowledge. However, other fanfics might focus more on other aspects of Naruto's character and not necessarily make him as smart as Minato.
There's a fanfic called 'Minato's Return'. In this story, Minato somehow returns and gets to see Naruto's growth. It's really interesting to see how he interacts with Kushina, who is also there in the story, and how they both guide Naruto in new ways. The descriptions of their family bond are heartwarming.
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.
In this fanfic idea, when Minato summons Future Naruto, it could also be for emotional reasons. Minato missed out on seeing Naruto grow up. By summoning Future Naruto, he gets to see the strong and capable man his son has become. Future Naruto can then tell Minato about his life, his friends, and all the adventures he's had, filling Minato with pride.
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.
The fanfic could portray Naruto as having greater physical strength and endurance. Maybe he can take more hits and keep fighting, while Minato might get tired more easily. Also, Naruto's strategic thinking in combat could be shown as superior, allowing him to outwit Minato in battles.
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.
Another character that could be important is Jiraiya. Since he was Minato's teacher and Naruto's mentor, he could play a role in the fanfic. Maybe he helps facilitate the meeting between Minato and Naruto, or he has some words of wisdom for both of them during their time together.
Well, in such a fanfic, there could be a training session where Minato teaches Naruto some of his signature jutsu. This would be a great opportunity for Naruto to learn more about his family's ninja techniques and heritage. It could also be a chance for Minato to see how strong his son has become.
In this kind of fanfic, a possible plot is that Minato and Naruto get to spend some normal family time together. They could go fishing, like Naruto used to do with Iruka. During this time, they bond and Minato gets to understand the simple joys that Naruto has in life, which he might have missed out on if he were still alive in the normal Naruto storyline.