Mesembryanthemum, if I understand your question correctly, is not 'just' a plant, but a whole Family of plants, including such genera as Lithops, Conophytum, Mestoklema, Trichodiadema, Ruschia and Pleiospilos, to name just a few. Is there a particular genus / species that you're asking about? This plant family, mostly succulent and mostly (?) hailing from South Africa, has hundreds if not thousands of members.

That's the trouble with common names, or at least one of the problems.

Delosperma can be called 'Ice Plant', and often is, but perhaps what they're referring to is a plant named Carpobrutus edulis, the ice plant one sees here on freeways, beaches and other open areas. Not all that pretty when compared with Delosperma (IMHO) but it's a tough plant.