Ok so first I want to say that I am not advocating animal cruelty. Tribbles are fictional and if you really think about it kind of creepy. My suggestion is for the Klingon side, as Klingons and Tribbles do not get along.
I suggest Klingons get to use Tribbles in Crafting. For example you have X amount of tribbles and a few other materials, you can make a speical Personal Armor set that gives you a permanent tribble buff. (about one third to a half of the standard tribble buff).
Obviously one could go the culinary route, but that's a little cliche.
Uh. They make pretty good non-scratching doorstops.
I'd like to be able to have them cosmetically infest my ship interior. Lying the the corner of a hallway every once in a while or something. Like being able to put an easter egg on your own ship for when you have guests over.