Often, Minato could be using his authority as Naruto's father or as a respected figure in the village. He might have made a promise to another clan or village and sees Naruto's marriage as the way to fulfill it. Another possibility is that he believes Naruto is not making good choices in his love life on his own and decides to take matters into his own hands to ensure Naruto has a stable relationship.
It could be a character from another important ninja clan, like Hinata perhaps. Since Hinata has always had feelings for Naruto and her clan is also a significant part of the village, Minato might see it as a good match.
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.
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.
He uses his skills as a ninja to protect Naruto. He sets up special seals around Naruto's crib to keep out any potential threats. Also, Minato might try to use his chakra to boost Naruto's weak body. He would transfer his own life force a little bit at a time to help Naruto fight the sickness, which is a huge sign of his love.
Well, in some fanfictions, Minato might use his powerful jutsu to break through the barriers between the living and the dead and directly reach Naruto. He could also try to influence the people around Naruto to bring him closer to where Minato is, like using his spirit to communicate with Naruto's friends and convince them to help.
Minato's presence can have a huge impact. He can act as a mentor, teaching Naruto things he never knew about his family and heritage. His wisdom and skills can inspire Naruto to reach new heights in his training and ninja development.
Minato might show his protectiveness by always being near Naruto in spirit. For example, he could leave behind some kind of jutsu or power that activates when Naruto is in great danger, like a hidden seal that releases a powerful defense.
Well, Minato might constantly be by Naruto's side in the fanfiction. He could prevent Naruto from taking on any dangerous missions, even if Naruto is capable. For example, he may interfere when Naruto tries to join a risky battle against a powerful enemy, saying it's too dangerous for him.
Naruto might initially be very confused and hurt. He loves his father and the village, so being banished would be a huge shock to him.
One common ability is their speed. Minato was known as the Yellow Flash for his incredible speed. Naruto, with his Sage Mode and later forms, can also move extremely fast. Another is their use of the Rasengan. Minato created the Rasengan, and Naruto has mastered it and even developed it further.