After breeding several full banks worth without obtaining anything much other than the 4 most common (grey, tan, brown, black) varieties, I'm starting to get more methodical in my experiments. I've been sticking 4 of the same color in the bank with 4 of the same type of food, and nothing else.
I'm a long way from drawing any solid conclusions yet, but it looks certain color/food combinations almost always yield a specific color offspring, with the occasional chance of a mutation. I've also sometimes seen tribbles consume food but produce no children, and sometimes decide they're not hungry. I also had one occasion where there were only a few tribbles in the bank and a stack of food, and the tribbles consumed almost the entire stack, producing an explosion of babies, in one reproduction cycle.
I suspect, but don't yet have enough evidence to back it up, that if you want the rarer tribbles, you need to feed them dropped food which can't be obtained from the replicator. The one uncommon (white) tribble I received came from a batch which included dropped food.
I just got my first one as a loot drop last night, threw it into the inventory and a few minutes later I had another. I think I'll keep the furry little guys around and see what kind of breeding they can do.
Is anyone looking into a tribble breeders guild? (not a fleet, just a loosley organized group dedicated to tribble breeding)