In this fanfiction concept, female Naruto might travel back in time due to some sort of time - space ninjutsu gone wrong or perhaps a special mission given by a powerful entity. When she arrives in the past, she quickly realizes the importance of saving Minato. She could form a strong bond with Minato, teaching him some of the advanced techniques she knows from the future, which in turn helps him in battles and ultimately saves his life.
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 possible storyline could be that Naruto discovers a powerful jutsu that allows him to travel back in time. He arrives at a crucial moment when Minato is in danger, perhaps during a battle against a powerful enemy. Naruto uses his skills and knowledge from the future to turn the tide of the battle and save Minato. Another idea is that Naruto's time travel is an accident, and he finds himself in the past with no way back immediately. He then has to figure out how to blend in while also protecting Minato from the shadows.
The story could start with female Naruto experiencing some strange phenomenon, like a bright light or a powerful jutsu backfiring, which then transports her back in time. As she lands in the past, she immediately recognizes the era and realizes that she has the chance to save Minato.
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.
Another idea is that Naruto's strong will and determination somehow reach Minato in the past. Minato, inspired by this unseen force which is actually Naruto from the future, finds the strength within himself to overcome a seemingly impossible situation. This could be during the Nine - Tails attack on Konoha, where Naruto's spirit gives Minato the idea to seal the Nine - Tails in a different way that doesn't cost him his life.
The village of Konoha would also be affected. With Minato still alive, he could continue to lead the village and make different decisions. This might change the relationships between Konoha and other villages. For example, there could be more peaceful alliances as Minato is known for his diplomatic skills. Also, Kushina being alive would mean that the Uzumaki clan might have a stronger presence in the village, which could lead to new jutsu developments and a different power balance within Konoha.
Well, Naruto has some similarities to Minato. Physically, they both have blond hair. In terms of abilities, Naruto's speed and some of his jutsu - like the Rasengan which was passed down from Minato - show a connection. Also, their leadership qualities are alike. Naruto became the Hokage, just like Minato was, and they both had to lead their village in difficult times.
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.