In this fanfiction concept, it could lead to a really interesting dynamic. Jiraiya is a powerful and eccentric ninja. He could teach Naruto some really unique jutsu, like more advanced forms of Rasengan. And their relationship would be filled with Jiraiya's boisterous personality clashing and also complementing Naruto's determination.
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.
Well, Naruto would probably learn a great deal more about the world of ninjas at a faster pace. Jiraiya was a very experienced and powerful ninja. He could teach Naruto some really advanced jutsu, like the Rasengan much quicker. Also, Naruto might grow up to be more like Jiraiya in terms of his attitude towards women, which could lead to some interesting adventures and misadventures.
One thing that could happen is that Naruto would learn a lot more about the toad sage mode earlier. Jiraiya could start training him right away. This would give Naruto an edge in battles. He would be able to use the toad - related jutsu more effectively and have a different combat style.
In fanfiction, Jiraiya's death can also be used to explore themes of sacrifice. He sacrificed himself for the greater good, and this can be further elaborated in different stories. For example, some fanfictions might show how his death inspired other characters to be more selfless.
Naruto is devastated. He feels a huge void in his heart. Jiraiya was the one who believed in him from the start, and his death makes Naruto question a lot of things. He might even start to doubt his own strength for a while.
One scenario could be that Jiraiya is poisoned before his fight with Pain. In the fanfiction, an enemy within the village who is secretly working for Pain's forces might have poisoned his food or drink. So, when he goes to face Pain, he is already weakened, and this ultimately leads to his death. Another possibility is that Jiraiya's death is a result of a time - traveling plot. If there is time - travel involved in the fanfiction, his actions in the past might accidentally create a situation where he has to die to preserve the timeline.
Naruto might also isolate himself in certain fanfic scenarios. He feels that no one can truly understand the depth of his grief. He spends time alone, reflecting on all the things Jiraiya taught him and the memories they shared, like their time training on that toad -infested island.