The story would change drastically. Naruto's identity as a descendant of Madara would immediately put him at odds with some in the village who still fear the Uchiha. His journey to become Hokage would be even more challenging as he has to prove that he's not going to follow in Madara's footsteps. Also, his training might take a different route. He could be taught some ancient Uchiha techniques by an unexpected mentor, perhaps an old Uchiha who has been in hiding.
He might become more brooding. The Uchiha are known for their intense emotions, and Naruto could inherit some of that. He may not be as openly optimistic as he was originally. Also, his sense of justice could be more influenced by the Uchiha's view of power and honor.
If Naruto were Madara's great - grandson in fanfiction, it would have a huge impact on the story. Naruto's place in the shinobi world would be redefined. His understanding of the ninja way would be influenced by both Uzumaki and Uchiha values. His interactions with the Hidden Leaf Village would be more complex. He could face discrimination from some who fear the power of the Uchiha, but also gain allies who see his potential. His relationship with Hinata could also be affected, as her family might view his new heritage in different ways. Moreover, Naruto's role in the overarching conflict between the villages and the forces of evil would be different, as he now has a different lineage to draw strength from.
One idea could be that Naruto has an innate ability to control the Sharingan more effectively than Sasuke. Since he's a descendant of Madara, his chakra might be more attuned to the Uchiha powers. This could lead to new jutsu combinations, like mixing the Rasengan with some form of Sharingan - enhanced fire style. It would also create a whole new dynamic in his relationship with Sasuke, as Sasuke would have to come to terms with Naruto having a stronger connection to the Uchiha heritage than he does.
Naruto might become more conflicted. Since Madara had some extreme views, Naruto would have to grapple with those ideas. He could become more secretive as he tries to understand Madara's motives better without alarming his friends. For instance, he might start training alone more often to study Madara - inspired techniques.
Naruto would likely become more serious. Madara's influence would probably make him less impulsive. For instance, he would think twice before charging into battle without a proper plan.
Their relationship would likely shift from being enemies to something more complex. Naruto's help might make Madara view him as an equal or even a friend. Madara, who has been alone in his ideology for so long, might start to respect Naruto's different perspective on peace.
Naruto would probably get Madara's Sharingan abilities. This could give him enhanced perception and the ability to copy jutsu more easily, like a supercharged version of his own ability. He might also gain access to Madara's Susanoo, which would be a huge boost to his combat capabilities.
He might have paler skin, like a typical vampire. His eyes could change color, maybe to a deep red. And his teeth could be a bit sharper, especially when he gets angry or hungry for blood.
The story would become much more intense. Naruto would suddenly have access to powers that were previously exclusive to the Uchiha. This would shift the power balance in the ninja world.
Her powers would likely be more focused on the Uchiha - related abilities. While Naruto has the Nine - Tails' chakra, she would have things like the Sharingan or even the Mangekyo Sharingan. She might be able to use genjutsu more effectively compared to Naruto.