Well, one possibility is that Naruto would grow up with more focused and intense training. Jiraiya could pass on his unique jutsu more directly. Maybe Naruto would learn about the world outside of Konoha in a more in - depth way earlier on, traveling with Jiraiya and having adventures that shape his understanding of the ninja world and his place in it.
The plot could involve Naruto developing a different personality. With Jiraiya as his adoptive father, he might be more confident from the start. Jiraiya could also teach Naruto about the importance of knowledge, like the secrets of the Sage Mode. And they might have to face enemies together that target Naruto specifically because of his status as the Jinchuriki, but with Jiraiya's guidance, Naruto becomes stronger to face them.
In this fanfiction, the story could show Naruto having a more stable family life compared to the original. Jiraiya could introduce Naruto to different cultures and people during their travels. Naruto would also get to know more about his parents through Jiraiya's stories. This would give Naruto a different perspective on his heritage. He might also form stronger bonds with other characters like Tsunade and Orochimaru in different ways since Jiraiya is connected to them.
Well, one possibility is that Naruto would become an even stronger ninja faster. Jiraiya could train him more intensively from a younger age. He'd also be exposed to more of Jiraiya's wild and carefree ways, which might make him more confident and a bit of a prankster like Jiraiya. And Naruto would probably have less of the loneliness he felt in the original story, since Jiraiya would be a more present father figure.
In a fanfiction where Naruto gets adopted by Jiraiya, Naruto's training would be intense and unique. Jiraiya has a lot of unorthodox methods. He could introduce Naruto to some very special and secretive forms of chakra control. Naruto might also learn more about the secrets of the Sage Mode much earlier. With Jiraiya's guidance, Naruto could form stronger bonds with other powerful ninjas that Jiraiya knew. This would give Naruto more allies and a greater understanding of the political and power dynamics in the ninja world.
In this fanfiction, perhaps Jiraiya didn't actually die. He was in a hidden place recovering from a serious injury. When he meets Naruto again, Naruto is shocked but overjoyed. Jiraiya then shares his newfound knowledge and powers with Naruto, and they team up to take on a powerful organization that threatens the Hidden Leaf Village. Jiraiya could teach Naruto some advanced Sage Mode techniques that he had been working on during his absence.
The plot might involve Naruto and Jiraiya getting transported to a different dimension. In this strange new world, they have to work together to find a way back home. They meet all kinds of strange creatures and characters. Jiraiya uses his wisdom and experience to navigate through this new environment, while Naruto's determination and growing power help them face the challenges. And during this adventure, Naruto might also develop a new form of his Rasengan, inspired by the things they encounter in this dimension.
One possible plot could be that Jiraiya is forced to betray Naruto due to some external threat. Maybe an enemy has captured Jiraiya's loved ones and threatened to kill them if he doesn't turn on Naruto. Jiraiya, in a difficult position, might have to pretend to be against Naruto to protect his loved ones while secretly trying to find a way to free both Naruto and his family from the enemy's clutches.
The plot could revolve around Naruto and Jiraiya going on a secret mission together. During this mission, Naruto discovers more about his family's past through Jiraiya's knowledge. They might encounter an ancient evil that was known to Naruto's parents, and Naruto has to use the skills he's learned from Jiraiya to combat it.
Well, one possible plot could be that Naruto is lost in a dangerous forest after a mission gone wrong. Jiraiya, who just happens to be passing by, discovers him. Jiraiya then decides to take Naruto under his wing for some special training. He might teach Naruto some new and powerful jutsu that Naruto hadn't learned at the academy. This could lead to Naruto becoming much stronger and more confident in his abilities.
Well, it could be about Naruto getting some really early and special training from Jiraiya. Maybe Jiraiya sees Naruto's potential right away and starts teaching him some basic but important ninja skills like how to better control his chakra. It might also involve some fun adventures where they encounter some small - time villains in the Land of Fire.
One possible plot could be that Tsunade and Jiraiya are deceived by false information about Naruto. Maybe they are made to believe that Naruto has turned to a dark side. For example, an enemy might have created false evidence of Naruto collaborating with a powerful evil force. Tsunade, being the Hokage, and Jiraiya, Naruto's mentor, would then be forced to take actions against Naruto that seem like betrayal from Naruto's perspective.
A possible plot could be that Naruto, under the care of Tsunade and Jiraiya, discovers new aspects of his heritage. Tsunade might know some secrets about the Uzumaki clan that she shares with Naruto. Jiraiya, being his godfather - like figure, helps him explore these new revelations. Meanwhile, the Akatsuki could still be a threat, and Naruto, with his new skills and knowledge, has to fight against them to protect the village and his loved ones.