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.
Minato might show his protectiveness by always being near Naruto in spirit. For example, he could leave behind some kind of jutsu or power that activates when Naruto is in great danger, like a hidden seal that releases a powerful defense.
At first, Naruto and Kushina might start to distance themselves from Minato. They could avoid him and not talk to him as much. As time goes on, the hatred might grow deeper. Naruto could start to oppose Minato in public or in important village matters. Kushina might also try to influence others in the village against Minato.
Minato being alive in fanfiction could mean more guidance and protection for Naruto. It could also lead to new alliances and conflicts within the ninja villages.
In fanfictions, Minato might cry for Naruto due to the missed opportunities. He missed Naruto's childhood, all those years of Naruto being alone. When Minato sees the kind of person Naruto has become despite all the hardships, it can be overwhelming and lead to him crying. He regrets not being there for his son.
One possible fanfiction could be about Minato and Kushina watching over Naruto from the afterlife as he grows into a great ninja. They could be his silent guardians, influencing his decisions in subtle ways.
Time - travel is also a frequent theme. For example, Minato might be brought to the present or Naruto sent back in time to meet his father. This allows for a lot of interesting interactions and character development. And of course, the theme of mentorship is there. Minato was a great shinobi, and fanfics often show him mentoring Naruto and passing on his skills and wisdom.
Well, 'Minato's Sorrow for Naruto' is another one. Here, Minato is brought back to life briefly and sees Naruto all grown up. He cries when he realizes how lonely Naruto has been without his parents. It's a touching exploration of their relationship through this unique scenario.
Another good one is 'Hidden Feelings'. In this fanfic, the author slowly builds up the relationship between Minato and Naruto. It starts with small interactions and gradually develops into something deeper, with great character development along the way.