Well, one possibility is that Naruto has some kind of premonition in the fanfiction. So he quickly rushes to Jiraiya's location where Pein is attacking. And he uses his shadow clones cleverly, distracting Pein long enough to rescue Jiraiya. He could also have some secret help from a new ally or an unexpected power boost from within himself.
In some fanfictions, Naruto might team up with other unexpected characters. Say, he teams up with some of the Akatsuki members who have a vendetta against Pein for some reason. Together, they are able to save Jiraiya from Pein's clutches.
Well, one way could be that Naruto accidentally stumbles upon Itachi's plot against Mikoto. Maybe Naruto was in the right place at the right time, and using his ninja skills and his never - give - up attitude, he manages to fend off Itachi and save Mikoto. Naruto is known for his quick thinking in dangerous situations.
One scenario could be that Naruto is poisoned during a battle. Jiraiya, with his vast knowledge of herbs and medicine, finds an antidote in time to save Naruto. Another possibility is that Naruto is trapped in a genjutsu created by a powerful enemy, and Jiraiya uses his own genjutsu - breaking skills to free Naruto.
One common plotline is Naruto arriving just in time to prevent Jiraiya from being fatally injured by Pain. He could use his newfound powers, like Sage Mode or even some advanced Rasengan variations. Another plotline might involve Naruto using his connection with the Nine - Tails to somehow sense Jiraiya's distress across great distances and rush to his aid. In some fanfictions, Naruto might also enlist the help of his friends like Kakashi or Sakura to form a rescue team for Jiraiya.
Pain is often one of the main antagonists. Since in the original story, Pain was the one who ultimately killed Jiraiya. So in fanfictions where Naruto saves Jiraiya, Pain is a natural choice to be the bad guy. Another common antagonist could be Orochimaru. He had his own vendetta against Jiraiya and the Hidden Leaf Village. He might be trying to use Jiraiya for some evil experiment or to gain access to the secrets of the Sage Mode.
He might miss out on some crucial training. Jiraiya has a lot of knowledge and skills to pass on, and by running away, Naruto could fall behind in his development as a ninja.
One possible reason could be that Naruto might be afraid of the intense training Jiraiya has in store for him. Jiraiya is known for his strict and grueling training methods, and Naruto, being young and sometimes impatient, might just want to escape that for a while.
In this kind of fanfiction, perhaps Itachi uses his quick reflexes and ninja skills to catch Sakura as she's about to fall. Maybe she was in a dangerous situation like on a crumbling cliff or a collapsing building. Itachi, being the powerful ninja he is, swoops in just in time to prevent her from getting hurt.
In this kind of fanfiction, Itachi might use his powerful Sharingan abilities. He could perhaps create illusions to mislead Orochimaru and his minions, then quickly snatch Sasuke away. For example, he could make Orochimaru think that Sasuke is in a completely different location while he secretly approaches and rescues him.