If you think of regular crystals as "space crystals" instead and that they have a regular structure that repeats in space then "time crystals" doesn't sound so awkward a term. Indeed that's what the principle investigator suggested was his inspiration... eg if Einstein said space and time are really "space-time" then could we have the "space-time" equivalent of crystals but repeating in time instead? At least that's how I'm reading the article.
Where I'm losing it... is that I never thought "space crystals" broke the symmetry of uniform space but instead that quantization seemed to me to be a function of the crystal (or atoms), not space itself.
However, taking the physicists enthusiasm at face value, this clearly appears (to me at least) to be another research avenue into unifying the space-time concept of general relativity with the incompatible space-time concept of quantum mechanics, and as far as I know the only one do-able in a laboratory. (Gravity wave experiments are detectors for events that happened "outside of a laboratory".) So... it's easy for me to share the enthusiasm even though I don't quite get it.