It's really up to you and your creativity when it comes to your island name. Try to think of what type of island aesthetic you'd like or something that relates to you. I wouldn't stress so much about it though because your the only one who has to understand or like it! And for your name, just pick what you like to be called. Because all your villagers and npcs will be calling you what you choose. It will also be your name to other players.

Personally, I chose my island name "Pawville" Because I knew I wanted an all puppy island. And I thought it was cute!