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.
Well, often in these fanfictions, Minato saves Naruto by using his powerful jutsu. He might use the Rasengan in a new and improved way to defeat an enemy that was about to harm Naruto. In a fanfic like 'Minato's Second Chance', he comes back to life with enhanced powers and uses them to shield Naruto from attacks. Also, Minato can save Naruto by teaching him things. In 'The Lessons of Minato', he imparts his knowledge to Naruto, which helps Naruto save himself in a dangerous situation, which is also a form of saving in a way.
One great 'Minato Saves Naruto' fanfiction is 'A Father's Promise'. In it, Minato manages to break free from the death seal's constraints just in time to save Naruto from a powerful enemy. Another one could be 'Minato's Return', where Minato is brought back to life through some powerful jutsu and he immediately focuses on protecting Naruto. And then there's 'Saving His Son', which shows a very emotional journey of Minato coming back to save Naruto from a life - threatening situation.
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.
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 emotional bond would be stronger. Naruto wouldn't have to face the pain of growing up without a father. Minato could be there to cheer him on during his training and be a source of comfort when Naruto faced difficulties, like when he was bullied as a child. He could also help Naruto better control the Nine - Tails' power from the start.
Another type could be a time - travel fanfiction where Naruto goes back in time and meets his father Minato. They could form a strong bond even before Naruto is born in the normal timeline. This might involve Naruto using his knowledge of the future to prevent certain tragedies from happening to Minato and the village. It could also show an emotional side as Minato gets to know his future son in a very different context.
A common theme is the exploration of their father - son relationship. Since Minato died when Naruto was born, fanfictions often delve into what could have been if they had more time together. For example, how Minato would have been as a father figure, guiding Naruto in his ninja training and in life in general.
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 interesting Minato and Naruto fanfiction could be about Minato surviving and training Naruto from a young age. This would change the whole dynamic of Naruto's growth as he'd have his father's direct guidance. Another might be a time - travel fanfiction where Naruto goes back in time and meets Minato before he was born, and they form a unique bond. There are also fanfictions that explore Minato's time in the afterlife and how he still influences Naruto's life.
Minato would probably be very proud. He might notice Naruto's use of his own jutsu and be impressed by how Naruto has developed his skills. Kushina would be more emotional, maybe crying at the sight of Naruto growing up so well.
One way it could happen is if Naruto somehow gains the ability to travel back in time. Maybe through a powerful jutsu or a unique kekkei genkai. He could then use his knowledge of future events and his combat skills to save his parents from their tragic fate.