With their high reproduction rate, tribbles have great potential for various other purposes than just being pets.
The tribble has been used in various cultures as a food and fur source, but with the wide-spread use of the replicator most Federation cultures have almost entirely done away with the usage of live animals for either. Also, the large number of breeding tribbles that would be kept on a fur or meat farm would create an enormous environmental hazard.
Tribbles have been used as test animals when poisoning was suspected. However, modern equipment can do the same task much better, for there are many substances which are poisonous to humanoids, yet harmless to tribbles.
Before the alliance between the Federation and the Klingon empire, tribbles were frequently used to sniff out spies. It was not unusual for Customs at a large starbase to have a tribble in addition to the sophisticated scanning systems. It is not unlikely that worlds outside the Federation, which have no treaty with the Klingons, do still use tribbles for this purpose.
Tribbles could, of course, be used as a weapon, when released in large numbers in an agricultural area. This is one of the reasons why the selection of breeders is strict, and why it is difficult to get a permit for transport of a breeding tribble. So far, the tribble has never been used in this manner.