What are the rarest bugs/fish for?

Solution 1:

When this question was posted, there was no use for such resources other than selling for bells. However, now there is a fairly new request type that allows you to give an animal any type of a specified category of resource (ANY fish or bug or fruit etc.). This gives the player a chance to utilize their rare items without requiring them from new players.

If you give a villager a rare item from these requests, you will typically get better rewards e.g. 1000+ bells and more resources than usual and/or a rare resource such as bronze treats or possibly even a fortune cookie.