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.
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.
In some versions, Minato might start by showing Naruto a simple demonstration. He could create a small mark and then suddenly disappear and reappear near it, catching Naruto's attention. Then he'd explain that Hiraishin is about using these special marks to instantly transport oneself to a marked location.
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.
Perhaps there was a curse or a disease that he couldn't overcome. In some fanfictions, they like to introduce unique and deadly diseases that not even the best medical ninja can cure. It could also be due to a failed mission where he was caught in a trap that was impossible to escape from, leading to his death.
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 possible story could be that Team Minato time travels to the future. They would be amazed by the technological advancements. Naruto might be excited to see how the village has grown. They could also learn new jutsu from the future ninjas and take that knowledge back to their own time to strengthen Konoha.
Well, it would probably shatter their relationship. They'd be filled with grief and guilt. Minato might blame himself for not being there to protect Naruto, and Kushina would be heartbroken. Their bond would be strained as they deal with this overwhelming loss, perhaps even leading to some estrangement as they struggle to cope in their own ways.
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.