How about a Klingonese Tribble Predator from the animated series episode, "More Tribbles, More Troubles"? Keep one in your inventory, and it eats non-exclusive regular tribbles that multiply in there. (Making sure that your rainbow HypnoTribble you got for pre-ordering remains uneaten.)
They should setup counters in Spacedock, DS9, Starbase 39, K7, etc that records how many tribbles get discarded there.
As the counter goes up tribbles start appearing in the scenery, the higher it goes, the more tribbles appear. Enough get discarded and the whole station goes into lockdown for a few hours and you get a special quest to help pack the tribbles away and send the to the new Tribble Prime the Federation has created, with a random quest reward that might include a rare tribble. (or like, a hypospray)