Commander Kruge kept a ferocious monster dog as a pet aboard his Bird-of-Prey to the left of his command chair. It died when two photon torpedoes from the USS Enterprise hit the ship. (Star Trek III: The Search for Spock)
The term "monster dog" is not the "official name" of this animal, however, it is consistently used by visual effects supervisor Ken Ralston during an interview in the Star Trek III: The Search for Spock Special Edition DVD, which discusses the creation of the puppet. According to magazine articles at the time of the release of the movie itself and, later, the DVD SE's text commentary by Denise and Mike Okuda, some of the members of the puppetry team dubbed it "Fifi Rebozo."
In the novelization of Star Trek III, the creature was named "Warrigul."
In the Universal Studios Hollywood theme park's Star Trek Adventure experience, a child participant would be selected to play a similar Klingon pet in each presentation. In the closing credits of the edited "episode," this juvenile creature was called a "puppy lizard."