Their relationship would likely become more equal. In the original story, Minato was more of a distant, heroic figure to Naruto. But through time travel, they can interact as peers, sharing knowledge and experiences. This would deepen their bond.
Well, perhaps a plot twist is that when they time travel, they end up switching bodies. Naruto, in Minato's body, has to deal with the responsibilities of being a Hokage in the past, while Minato, in Naruto's body, has to navigate through Naruto's life in the present. This would create a lot of comical and also very tense situations as they both try to adapt and not mess up the timeline.
In a Naruto time - travel Minato fanfiction, Minato could be sent to the future. There, he would see how the village has changed, and might be shocked to find out about the Akatsuki and their actions. He could then work with Naruto and the others to fight against the Akatsuki using his unique skills and knowledge of the past. For example, he could teach Naruto some new jutsu that was lost over time.
If Minato time - travels, it could change a lot. For instance, if he prevents the Nine - Tails attack, Naruto would grow up in a very different environment. He might not be the underdog he was in the original story. Also, if Minato changes things during the Uchiha clan's situation, Sasuke's story would be completely different. Konoha's political situation would also be altered. There would be less animosity between different clans perhaps.
It would be very close. Naruto has always longed to know his father, so when he travels back in time to meet Minato, they would quickly bond. They'd probably start training together right away.
Jiraiya is likely to be a popular character. He was Minato's teacher and has a strong influence on Naruto as well. In a time travel fanfiction, he could provide a lot of wisdom and help in the different situations they encounter. Also, Tsunade could be important. Her leadership skills and medical ninjutsu could be very useful when they are in the past, and she has her own history with both Minato and Naruto's family.
One great 'Minato x Naruto Time Travel Fanfiction' is 'A Father's Chance'. In this fic, Minato travels back in time to Naruto's childhood and gets to be the father he always wanted to be. It's really heartwarming to see their interactions.
One possibility is that Naruto's presence in the past would disrupt the timeline significantly. However, it could also lead to a strengthening of Konoha. He and Minato could work together to improve Konoha's defenses and train the next generation of ninjas earlier. This could potentially change the entire future of the ninja world, making it a more peaceful place with Naruto's knowledge from the future.
One possible scenario in this fanfiction could be that female Naruto, with her knowledge of future events, is able to prevent Minato's death. Maybe she uses her skills and the element of surprise to change the course of the battle where Minato was supposed to sacrifice himself. She could also have some unique abilities or jutsu that she uses to outwit the enemies and keep Minato alive.
One interesting plot could be Minato traveling back to prevent the Nine - Tails attack on Konoha. He could use his knowledge of future events to stop Orochimaru from influencing the situation that led to the attack. Another plot might be Minato traveling to a time when the Uchiha clan was on the verge of the coup. With his wisdom and power, he could try to find a peaceful solution and prevent the massacre. Also, he could travel to the time of the First Hokage to learn more about the origin of the ninja world and bring back that knowledge to improve Konoha in his own time.