Well, in a fanfiction where Naruto travels to the past, he could meet his parents when they were young. He could also learn new jutsu from the past ninjas that were lost over time. Maybe he'd even be able to influence the early development of the Hidden Leaf Village in ways that would make it stronger in his own time.
Naruto would meet his younger self. This would be a really strange situation. He could also meet all the important shinobi from the past. For example, the First and Second Hokage. He might have a chance to learn from them directly, instead of just hearing about them in stories. Additionally, he could meet some of the characters who were only mentioned in passing in the original story but had a big influence in the past.
In such a fanfiction, Naruto might try to prevent some of the tragedies that occurred in the past. For example, he could stop the Uchiha massacre. He would use his knowledge from the future to change things. Maybe he'd also be able to save some characters who died early in the original story, like Jiraiya. And he could potentially have a huge impact on the relationships between different clans, as he'd know about future conflicts.
One interesting plot could be Naruto using his future knowledge to prevent the Uchiha massacre. He might try to change Itachi's situation and find a way to resolve the conflict between the Uchiha clan and the village without all that bloodshed.
Jiraiya. Naruto could have a different kind of apprenticeship with Jiraiya. He could share some of his future experiences with Jiraiya, which might change how Jiraiya trains him and also how Jiraiya approaches his own missions and prophecies.
Naruto's character could develop in many different ways. If he travels to the past, he'd have to adapt to the different ways of the old ninja world. He might become more respectful of the traditions as he sees how they were formed. Also, he could become more of a mentor figure to the younger versions of his friends, which would change his self - perception from a student to a teacher, thus making him more mature and responsible.
Naruto's character might develop a greater sense of responsibility as he realizes the impact of his actions in the past on the future. He may become more patient, knowing that he has to tread carefully to avoid major disruptions.
He might accidentally change the future. For example, he could prevent important events from occurring that were crucial for the development of the present he came from. Maybe he meets his parents at a younger age and reveals too much about the future, which could cause them to make different decisions.
One great fanfiction could be where Bella travels back and she and Jasper form an unexpected alliance against the Volturi. They use their combined knowledge of the future and Jasper's combat skills to change the fate of the Cullen family.
Another possible main character could be a historian or a scholar within the order. Their knowledge of the past could be crucial in helping the order navigate the historical period they travel to, and they might also be on a personal quest to discover something that has been lost in time.