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.
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 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.
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.
Kushina could be an important character even though she's dead in the original story. In this fanfic, her spirit or memories could play a role in Naruto's upbringing. For example, Minato could tell Naruto stories about Kushina's feisty personality and her love for the village.
In some fanfictions, Minato saves Naruto physically. For example, he might use his teleportation jutsu to appear right in the middle of a fight and whisk Naruto away from danger. In others, he could save Naruto spiritually. He might enter Naruto's mind, like in the fanfic 'Minato's Mind - Saving Mission', and give him the strength and confidence to overcome his inner demons. Sometimes, Minato saves Naruto from a political situation. For instance, in 'Minato and the Hidden Leaf's Politics', he uses his influence and wisdom to protect Naruto from being used as a political pawn.
Minato would likely be extremely shocked. Naruto is his son, and the idea of Naruto attacking him would be completely unexpected. He would probably try to talk to Naruto first, to find out what is wrong and why he is being attacked.