The Straw Hat crew would probably be very surprised at first. Nami might worry about how Hancock's presence would affect their finances, since Hancock is used to a more luxurious lifestyle. Usopp would be in awe of Hancock's beauty and powers. Chopper would be excited to have Hancock around, as she could potentially teach him about different kinds of medicine from Amazon Lily. Zoro might be a bit skeptical at first but would come to accept it as long as Luffy was happy.
Well, in this fanfiction, the Straw Hats might be in need of a powerful ally. They meet Hancock, and she is instantly drawn to Luffy. The crew could then face a common enemy that is threatening both Amazon Lily and their own goals. Hancock decides to join forces with the Straw Hats. Sanji would be over the moon to have Hancock around, but also a bit jealous of Luffy's special connection with her. And Zoro would be his usual stoic self, just glad to have extra muscle for the upcoming battles.
Luffy could be the main character. Since Hancock has a special connection with him, the story could revolve around Luffy's interactions with Hancock and how he manages to get the Straw Hats and Hancock's people to work together.
One possibility is that they gain a deeper understanding of Luffy's motives. For example, they might see his memories of making the promise to Shanks and realize how important it is for him to become the Pirate King. This could strengthen their resolve to support him even more.
Luffy might be excited to have a new member, as he always is about making new friends. But he's also not stupid, he would keep an eye on Kuro. Zoro would be very skeptical. He doesn't trust easily, especially someone with Kuro's reputation. Nami would probably try to figure out how she can benefit from Kuro's skills, while also being cautious not to be double - crossed.
The relationships will likely become more profound. For instance, Zoro might see Luffy's memories of believing in his strength even when others doubted, which could make Zoro more determined to protect Luffy at all costs.
It could be Chopper. Since Chopper is a doctor and Reiju has some medical knowledge, they might bond over sharing medical techniques and treating the crew's injuries. Also, Chopper's kind - hearted nature would make him a great friend for Reiju who might be used to a more cut - throat family environment.
If Naruto was raised by the Straw Hats, he'd probably be a very different person. He might be more carefree like Luffy. For example, he'd be used to the wacky adventures on the Grand Line. His ninja skills could blend with the pirates' combat styles, creating some really unique fighting techniques.
Well, as of the latest manga chapters, Yamato has indeed become a part of the Straw Hats. This decision brings new dynamics and adventures to the crew.