What makes a good creature design

Creature design is a wonderful subject. Few artists didn’t attempt at least once to make a creature design work. Be that a hybrid, a whole new creature, or maybe a design that roots into some form of technology. But what makes a good creature design?
Surrounded by fauns and mermaids, chimera and centaurs, one may think ‘oh, creature design is easy, we take one part, and add another animal part, maybe even a third and we’re done. But it isn’t that simple. Sure, we can make wonderful creature designs by cutting and pasting animal parts, but true creature design is FAR more than that.