In a fanfiction where Naruto is Tsunade's son, his training regime would be very different. Tsunade would likely push him hard in both physical and medical training. Naruto might have access to resources that he didn't have in the original story. For example, he could study ancient Senju clan jutsu from a young age. His appearance might also be different, perhaps having some of Tsunade's features. His hair color or eye shape could be influenced by her genetics. Moreover, the village's perception of him would change. People would see him as not just the Jinchuriki of the Nine - Tails, but also as Tsunade's heir, which could bring both advantages and disadvantages in terms of how they interact with him.
Well, Naruto would probably have a different upbringing. Tsunade could teach him her gambling skills, for better or for worse. He might also be more confident in his abilities from the start, since Tsunade is a very self - assured person. And he could have a different relationship with Jiraiya. Maybe Jiraiya would be more cautious around Naruto, knowing that Tsunade is his mother.
One possible storyline could be Naruto growing up with a different set of values influenced by Tobirama's more strict and strategic mindset. He might be more of a tactical thinker from the start. For example, in battles, he could quickly analyze the opponent's jutsu and come up with counter - strategies more efficiently than in the original series.
Naruto's character might develop a more diplomatic side. The Senju were often involved in negotiations and maintaining peace. So, he could be more inclined to talk things out rather than jump into battle right away. He would still have his energetic and determined side, but with a touch of Senju wisdom. For example, when dealing with other villages or clans, he would use his Senju heritage to build bridges and find common ground rather than just relying on his strength as a ninja.
Naruto might be less impulsive. Tobirama was a very calculated person, so Naruto could inherit that trait and think twice before acting. For instance, instead of rushing headlong into battles, he would plan his moves carefully.
Sure. There are some out there. The idea of Naruto being related to the Senju clan opens up a whole new realm of possibilities in the Naruto fanfiction world. For example, it could change his relationships with other characters. If he was Senju, his connection to Sasuke, who is an Uchiha, would have even more depth considering the historical rivalry and connection between the Senju and Uchiha. And it might also give Naruto access to different jutsus and abilities that are unique to the Senju.
Well, a plot could involve Naruto facing challenges related to his new identity as a Senju. There could be those who oppose him having the Senju blood, perhaps out of jealousy or fear of his growing power. He might have to prove himself worthy of the Senju name. Also, his friendship with Sasuke could take on a new dimension. Since Sasuke is an Uchiha, there could be an interesting dynamic between the two clans' legacies within their friendship. Naruto could use his Senju powers to try and bridge the gap between the Uchiha and the rest of the village, as the Senju were often associated with peace - building in the past.
Well, if Naruto was Hashirama's son, the power dynamics in the Naruto universe would be completely different. Naruto might have inherited Hashirama's Wood Style from the start, which is an extremely powerful ability. This could have made him a target much earlier in life, as many would fear his potential. Also, his relationship with Sasuke might have taken a different turn. Since Hashirama and Madara had a complex relationship, Naruto's connection to Hashirama could have added more layers to the Naruto - Sasuke dynamic.
Yes, there are likely fanfictions with that theme. You can check popular fanfiction websites like FanFiction.net or Archive of Our Own. Just search for 'Naruto is a Senju Graduation' on their search bars and you might find some interesting reads.