Well, one possibility is that Naruto could learn a great deal more about the past of the ninja world directly from Jiraiya. He might also get to witness some of the key events that shaped the present he came from. For example, he could be there when Jiraiya was gathering information on the Akatsuki, which would give him a huge advantage in dealing with them in his own time.
His character would likely become more mature faster. Being in the Jiraiya era, he'd see the harsher realities of the ninja world at an earlier stage. For instance, he might witness more battles and political intrigue up close, which would force him to grow up quickly.
In time travel fanfiction, if Jiraiya goes back, it can have a huge impact on his relationships. He may be able to save some relationships that were lost in the original timeline. For instance, he could repair his broken friendship with Orochimaru. Regarding his relationship with Naruto, if he travels back to Naruto's childhood, he can be a more effective teacher and friend. This could lead to Naruto trusting him even more. And if they travel back together, they rely on each other more in the different time period, which enhances their bond.
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.
One popular plot is Jiraiya going back in time and using his knowledge to prevent the Uchiha massacre. He could train Sasuke differently, or even find a way to resolve the conflict between the Uchiha and the village. Another plot might be Naruto traveling back with Jiraiya. They could team up to change events that led to the rise of Akatsuki, like stopping Nagato from going down the dark path. There are also plots where Jiraiya tries to change his own past mistakes, such as not being able to save Orochimaru from the darkness earlier.
Jiraiya would likely become more protective. He has always been a bit of a free spirit, but with a son in the picture due to time travel, he would be more cautious. His focus would shift from just his own adventures and spying to ensuring his son's safety and growth.
Jiraiya time travel fanfiction can be really exciting. It allows for so many interesting plot twists. For example, Jiraiya could go back in time and prevent certain tragedies in the Naruto universe. He might be able to save his friends or change the course of important battles. This could lead to a completely different story arc, with new alliances and enemies emerging.
A possible plot in Jiraiya time travel fanfiction could be Jiraiya traveling back to the time when the Uchiha clan was still intact. He could try to prevent the massacre, which would have a huge impact on the Naruto story. This would change the relationship between Sasuke and Naruto, as Sasuke's motives for revenge would no longer exist.
It could be Orochimaru. Since he has a history with Jiraiya, he might target Naruto while he's under Jiraiya's tutelage, seeing Naruto as a potential threat in the future. Orochimaru could try to steal Naruto's power or use him in some dark experiments.
Well, in this fanfiction scenario, perhaps Jiraiya's time travel is an accident. His son could be a half - breed, part - something special like a jinchuriki of a new tailed beast. Jiraiya then has to use his knowledge from the past and new skills he learns in this timeline to protect his son and the village from those who want to use the son's power for evil.