This is such a cool idea! I love pretty much every idea people have added to it in this thread too. I odn't think it's out of place in AC - we have beaches and islands, what's so different about caves?
I had a cool idea for the next installment of Animal Crossing: spelunking.
There could be a cave or caves that your character could explore. Perhaps you could use your shovel to dig in certain spots for minerals, or a new tool like a pickax or something to dig in the cave walls for valuable items/collectibles. There could also be special bugs and fish that could be catchable in the caves, i.e. blind cave fish.