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.
If adopted by Jiraiya, Naruto would probably be more self - assured. Jiraiya would instill in him a sense of pride in his abilities. He'd also be more of a free spirit. Jiraiya's influence would mean Naruto would be more likely to take risks and explore new things without as much hesitation as in the original story. He might also be a bit more flirty, as Jiraiya has his own charm when it comes to the opposite sex.
Jiraiya might comfort Naruto by sharing his own experiences. He could tell Naruto about the times he faced difficulties and how he overcame them. For example, he might talk about his training with the toads and how he grew stronger through perseverance.
In fanfiction, Jiraiya seeing Naruto as Hokage can be a very emotional moment. It could be a vision Jiraiya has in a story where he's still alive, or perhaps it's a memory in a story set after his death. This often shows Jiraiya's pride in Naruto and his belief in Naruto's potential to lead the village.
Their relationship would be very strong. Jiraiya would see Naruto as not just a student, but also as a connection to Minato. He would be strict but also very affectionate. Naruto would look up to Jiraiya as a father figure, since he never really had one. They would share many adventures together, with Jiraiya teaching Naruto about life, love, and ninjutsu.
One possible reason could be that in this fanfiction, Jiraiya might be preoccupied with other important matters. Maybe he has a secret mission that demands his full attention and he simply doesn't have the time to train Naruto. Another reason could be that there is some sort of conflict or misunderstanding between them that prevents Jiraiya from taking Naruto under his wing. For example, Naruto might have unknowingly offended Jiraiya in some way, or there could be a prophecy or external force that is keeping them apart for the time being.
In the Naruto universe, bringing someone back from the dead is extremely difficult. While there are some jutsu like the Edo Tensei that can reanimate the dead, it has its limitations and is often considered dark magic. For Naruto to revive Jiraiya, he would need to find a way around the natural laws of life and death. However, in fanfiction, anything is possible as it allows for creative interpretations and bending of the established rules.
Jiraiya's reaction might be complex. He might be in awe of Naruto's new power but also sad about his departure. He could see it as a natural progression in Naruto's journey, yet still feel a pang of loneliness. He might start to reflect on his own life and training methods, wondering if he could have done something differently to keep Naruto with him. At the same time, he would be eager to see what Naruto will achieve in the world with his newfound abilities.
They often make him seem irresponsible. For example, in normal Naruto story, Jiraiya is seen as a mentor to Naruto. But in bashing fanfiction, he may be shown as being more interested in his own adventures and leaving Naruto alone at crucial times.
It could be Orochimaru. Since Orochimaru has a long - standing feud with Jiraiya and is always up to no good. If Naruto ditches Jiraiya, Orochimaru might see this as an opportunity to manipulate Naruto. He could try to convince Naruto that Jiraiya was holding him back and that he can offer Naruto more power, all while having his own evil plans for Naruto.