Based on spending lots of time camping rares on the island (got 7/11 pets), there are two categories of rare spawns:
Timed ones, all on a 30-60 minute timer:
Bufo, Gu'chi, skunky brew event, Spirit of Jadefire, Leafmender, Garnia.
The best way to get these is to note down the time when they last died, show up half an hour later and then camp until they appear. If you time it right, they'll always appear within half an hour of you showing up.
Spawned ones, all having a chance to spawn instead of a random animal of their type:
Monstrous Crawler, Imperial Python.
You (or someone else) has to kill crabs or snakes to get these two to spawn. The chance of them appearing is pretty bad, so expect this to take a while. Note that there are crabs in more than one place on the island.