Are you on the riverland farm? Since you're talking about various islands.
I'm on year 5, recently caught all the fish (this is after coming to this forum explicitly to beg for help because I could not catch an eel). This is the furthest I've gotten on a save file. I have 48 very happy rabbits! I need more names if I'm going to get more.
Rabbit names: villager names always work, but putting “lil” in front of the name is even better. “Lil Gus”, “Lil Marnie”, etc. Also cereal mascots, like Trix or Tony or Cap. Crunch. Then mythical creatures like Cthulhu the bunny. Then characters from a book or movie you like, but slightly altered. Hairy Potter, Frodoe Baggens, etc. Then there’s always Greek Gods, like Hera, Persephone, Eros, etc. and also Toasty.