My island is named Poveglia, which is the name of an actual island in the Venetian Lagoon of Italy. The actual island of Poveglia was used as a quarantine station for people suffering from the bubonic plague, then later housed an asylum and a nursing home, before being abandoned. It is supposedly haunted, and I first heard of it like two decades ago due to that reason (it was featured on a tv show).
I chose the name for a few reasons. One reason is that when New Horizons came out, we were first being flung into the COVID-19 being a global pandemic, so pandemics and quarantine were on the mind. Another reason is that I've always been interested in horror, spooky stories, ghosts and paranormal stuff (though I don't really believe in ghosts or whatnot, it's always just been kinda fun to like read or hear people's stories) so I kinda wanted to give the island a "spooky" name, like how my main New Leaf town is called Haunton. Additionally, since the actual Poveglia has long been abandoned, it somewhat fit the "deserted island" theme of the game. The big reason, though, is that no other names were coming to mind and, for the previously stated reasons, when Poveglia came to mind I just kind of went with it.
If there were an option to change the name, I probably would since nowadays I feel like I would prefer a wholly original name. I've seen other Poveglias and, while that's fine and understandable since it's a known place, it also just sort of feels awkward. There's also the possibility that eventually the actual Poveglia will lose its appeal as a creepy, supposedly haunted location because there have been a few plans over the years to develop the island, though none have thus far been successful. All that said, since there is no such option to change the name and there likely never will be, and also because I've made far too much progress in the game to ever want to restart or anything of the sort, I am content with what I've got. It still has a nice ring to it and is fun to say, at least.