In this type of fanfiction, it's possible that the story delves into Naruto's backstory and how being Sarutobi's son would have affected his childhood. Maybe he would have grown up with more guidance and support, which could have changed his personality. Instead of being the lonely outcast that he was in the original story, he could have been more confident from the start. The fanfiction might also introduce new characters related to Sarutobi that play important roles in Naruto's life. There could be new adventures and challenges that are unique to this new family dynamic, such as Naruto having to uphold the Sarutobi family name and legacy within the Hidden Leaf Village.
Obviously, Naruto would be one of the main characters. And since it's about him being the son of Sarutobi, Sarutobi himself would be a crucial character. There would also likely be the usual Naruto characters like Sasuke, Sakura, Kakashi etc. as they would still be part of Naruto's life in the fanfiction, even with this new family relationship.
Well, a plot twist could be that since Naruto is Hiruzen's son, the Akatsuki targets him differently. Instead of going after the Nine - Tails inside him in the usual way, they might try to use his relationship with Hiruzen against him. They could try to blackmail Hiruzen into making certain concessions for the safety of his son. Also, Naruto's training might be more intense and different. He might be taught more about the political side of being a ninja by Hiruzen himself, which changes his perspective on the ninja world.
Perhaps in this fanfiction, Sarutobi is trapped in a genjutsu created by a rogue ninja. Naruto, being highly resistant to genjutsu due to his Uzumaki blood, manages to break into the genjutsu world. There, he has to fight his own inner demons as well as the rogue ninja's illusions to free Sarutobi.
I really like 'Naruto, Sasuke and Sarutobi: A New Beginning'. In this fanfic, it starts with Sarutobi having a premonition about the future of Naruto and Sasuke. So he takes steps to guide them differently. It's an interesting take on their relationship. 'Sasuke's Redemption under Sarutobi' is also good. It shows Sasuke trying to make amends with the village with Sarutobi's guiding spirit in mind. And 'The Hidden Power of the Trio' is a fanfic that gives a new perspective on the powers they possess and how they can be combined for the greater good.
If Naruto were Hiruzen's son in a fanfiction, his character development would take a whole new turn. He'd grow up with a different set of values instilled in him. He might be more diplomatic from the start, having been exposed to Hiruzen's leadership style. His motivation to become Hokage might be less about seeking recognition and more about following in his father's footsteps. Also, his relationship with the village elders would be different. Instead of being an outcast, he'd be seen as the heir, which would bring its own set of challenges and opportunities for growth.
Perhaps he uses some ninja techniques like a shadow clone jutsu. He could make multiple clones to help him sort through and complete the paperwork quickly.
Naruto would probably be extremely hurt. He has always looked up to Sarutobi as a leader and a father figure. His first reaction might be disbelief, unable to accept that Sarutobi would do such a thing. Then he would likely feel a great sense of anger and betrayal. He might lash out and try to prove his innocence.
One of the popular characters paired with Sarutobi is Jiraiya. They are both important figures in the Hidden Leaf and their relationship, whether it's as comrades or as teacher - student in some fanfic interpretations, is always interesting to explore.