Well, if we think about it, in this fanfiction Naruto as Odin's son would likely have some unique abilities. He might be able to use some Asgardian magic along with his chakra. When it comes to his relationship with Natasha, it could be a friendship based on mutual respect for each other's combat abilities. They could learn from one another, like Naruto teaching Natasha some of his ninja stealth techniques and Natasha showing Naruto how to handle modern weapons. It would be a cool dynamic to explore in the fanfiction.
The plot might involve Naruto arriving in Natasha's world by accident. Since he is Odin's son, he has a lot of power but also a lot of responsibility. Natasha is intrigued by this strange newcomer with such unique abilities. They start to investigate a series of strange events that seem to be linked to some ancient Asgardian magic. As they dig deeper, they find out that there is a villain who wants to use Naruto's connection to Odin to gain ultimate power. So they have to fight this villain together while also dealing with their own personal growth and the relationship between them.
His friends from the Leaf Village like Sasuke and Sakura could be his allies. They have always been by his side in the original story and would likely support him even with this new twist.
The plot might involve Naruto being a son of Odin and having to deal with the expectations and responsibilities that come with such a heritage. He could be caught between the ninja world he grew up in and the world of Norse mythology. For example, he may need to reconcile the values of the shinobi with the values of Asgard. There could also be a conflict where he has to face off against enemies who are after the power he inherited from Odin. This could lead to him going on a journey to learn more about his true father and the powers he possesses, all while still trying to keep his identity as a ninja intact.
In this fanfiction concept, it could be that Odin had a secret mission in the Naruto world. He might have fallen in love with a kunoichi there, perhaps someone with special powers or a noble lineage. Their union could have led to Naruto's birth. It would also mean that Naruto would potentially have some latent god - like powers from Odin that could be explored in the story, like enhanced strength or the ability to control some Norse - inspired elements.
He could have interesting interactions with Issei. Naruto's more selfless nature compared to Issei's pervy tendencies could lead to some humorous and also teaching moments. Maybe Naruto would try to get Issei to focus more on noble goals rather than just chasing girls.
He might have the power of Odin's All - Father magic. This could allow him to see the past, present, and future, just like Odin's ability to gain wisdom through his ravens. Also, he could potentially have enhanced physical strength, much greater than what he has as a regular ninja, allowing him to overpower even the toughest opponents with ease.
In most traditional mythologies, Percy Jackson is the son of Poseidon in the 'Percy Jackson' series, and Odin is from Norse mythology. But in fanfiction, anything can be reimagined. So, he could be depicted as the son of Odin in some very creative fan - made stories.
The sons of Odin included Tyr, Heimdallr, Vidal, Wally, Hermoder, Blaki, and others. Tyr was the god of justice in Norse mythology. He symbolized justice, vows, and courage. He was in charge of war and law. He lost an arm to prove his bravery. Heimdallr was known as the patron saint. He could see and hear from afar. He always rode a golden mane horse and carried a war horn on his back, guarding the Bifröst at the entrance of Asgard day and night. Vidal was known as the "God of the Forest" or the "God of Silence" and was regarded as an indestructible force of nature. During the Twilight of the Gods, he tore the giant wolf Fenrir in half, and it was rumored that he lost a leg because of it. Wally grew up on the first day he was born. Without washing his hair, he killed Hodel with a bow and arrow to avenge Bader, and later became the "God of Gardening." Hermoder was good at flying. He was Odin's special attendant, responsible for running errands. He was also the messenger of the gods. He was in charge of Odin's Gungnir (Spear of Eternity). He was the only other person who could ride the Eight-legged Divine Steed. Brachi was the God of Poetry and Music. There was also Bard, Thor's brother.
His relationship with the Avengers would be different. They might see him as Natasha's legacy. For example, Clint might take on a more protective role towards Peter, seeing him as a part of Natasha that still lives on. And Tony could be more involved in mentoring Peter, not just because of his own interest but also because of his respect for Natasha.
I'm not sure there are many official crossovers like this. However, one could imagine Naruto's ninja skills being quite a shock to the people of Wakanda. He might be intrigued by their advanced technology, while Natasha could find Naruto's chakra abilities very different from the superpowers she's used to in the Marvel universe. Maybe Naruto would learn from Wakanda's warriors and Natasha could learn some of Naruto's stealth techniques.