The story would likely be very different. If the Shinigami is female and related to Naruto's father, there could be a whole new backstory. Maybe Naruto's father had to make a sacrifice to the Shinigami for some reason. This could affect Naruto's own journey as he might inherit some of the consequences of that relationship.
His powers would likely include some form of soul manipulation. He might be able to sense the souls of others more acutely, which could give him an edge in battle by predicting his opponents' moves.
He'd probably be more solemn. The Shinigami is a being associated with death, so Naruto would be exposed to serious concepts early on.
His story would be completely different. Naruto would likely grow up with a different set of values as the son of the Shinigami. He might be more in tune with the balance of life and death, rather than just focusing on being accepted in the village. His relationships with characters like Kurama would also be different. Since he has a connection to the Shinigami, Kurama might be more cautious around him or even see him as a potential threat to its own existence.
In fanfiction, the possibilities for Naruto's father could be many. It could be an original character created by the author. Maybe a powerful shinobi who has a connection with the female Shinigami. For example, he could be a shinobi who made a deal with the Shinigami in the past and through some convoluted plot, became Naruto's father.
Well, in fanfiction, the interaction could be really complex. Maybe the Rinnegan has a power that can bind the Shinigami to Naruto's will. But at the same time, the Shinigami might try to corrupt Naruto because of the power of the Rinnegan. Naruto would have to resist this corruption while still using the Shinigami's power to fight against his enemies. He could also use his Rinnegan to unlock secrets about the Shinigami's past, which could help him in his battles or in making important decisions about the ninja world.
Naruto would probably be more independent in his thinking. Instead of relying on the typical ninja values taught in the village, he'd draw from the Shinigami's teachings. This could make him a bit of a rebel in the eyes of the ninja community. He might question the established norms more often. His development would also be influenced by any special tasks or missions the Shinigami gives him. For instance, if he has to deal with restless souls, it would make him more empathetic towards those in pain, but also more determined to maintain the order between the living and the dead.
His speed and strength could increase. Shinigami powers might give him a boost in physical abilities as they are often associated with otherworldly strength.
I'm not sure specifically as there could be many different interpretations in fanfiction. It might be about Naruto having some sort of special connection to kami (gods) and shinigami (god of death) in a made - up scenario, perhaps involving him being related to them in an unexpected way.
His relationship with his friends would likely change. He might become more distant at first as he focuses on his training with the female shinigami. But in the long run, his new powers and insights could make him a better protector of his friends. For example, he could use his newfound skills to rescue them from more dangerous situations.
Well, in some fanfictions, the Shinigami might save Naruto by using its special powers to reverse a fatal curse or injury. For example, it could extract a harmful entity that was about to kill Naruto from his body.