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.
A possible storyline could involve Naruto's time - traveling being a result of some sort of temporal distortion caused by a powerful enemy's jutsu. When Naruto gets to the past, he discovers that Minato is in a situation where he's outnumbered and outmatched. Naruto steps in, using his Nine - Tails chakra mode which is far more advanced than what anyone in the past has seen. He scares off the enemies and saves Minato. After that, Naruto has to be careful not to change too much of the past, but also tries to find a way to return to his own time while making sure Minato stays alive.
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.
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.
One possible storyline could be Minato's struggle to find a cure for Naruto's sickness. He might search through all the hidden jutsu in Konoha, asking for help from the elders and the medical ninjas. Another could be Minato's emotional journey as he takes care of his sick baby. He'd be filled with worry, but also determination to see Naruto healthy. For example, he could sacrifice his own sleep to stay by Naruto's side, constantly monitoring his condition.
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.
One possible storyline could be Naruto growing up feeling isolated and constantly seeking validation. He might strive harder in his ninja training to prove himself, not just to the village but also to the memory of his father. Another could be that he forms stronger bonds with other outcasts in the village, like Rock Lee, who also had to overcome challenges.
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.
One interesting plot could be Minato traveling back to prevent the Nine - Tails attack on Konoha. He could use his knowledge of future events to stop Orochimaru from influencing the situation that led to the attack. Another plot might be Minato traveling to a time when the Uchiha clan was on the verge of the coup. With his wisdom and power, he could try to find a peaceful solution and prevent the massacre. Also, he could travel to the time of the First Hokage to learn more about the origin of the ninja world and bring back that knowledge to improve Konoha in his own time.
One interesting plot could be Minato and Naruto going back in time and Minato getting to see Naruto grow up from the start. He could teach Naruto things he didn't get to in the original timeline, like more advanced jutsu. Another plot might involve them changing key events in the past that had a major impact on the future of the ninja world. For example, preventing certain tragedies or battles.
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.