I agree that the general public deserves to have more common names for bee genera at its disposal. It would also greatly help young students and amateur naturalists.
I would not suggest the common name "stone bee" for Lithurgus, although the genus name is derived from the ancient Greek lithos ("stone").
In German-speaking countries, the genus is called the Steinbiene (stone bee).
This is quite unfortunate since the two middle European species (Lithurgus chrysurus and Lithurgus fuscipennis), and as far as I know all other species, nest in wood or hollow twig.
It was Malyshev (1930 Nistgewohnheiten der Steinbienen, Lithurgus) who surmised that they owe their name to the almost bare and shining surface of the mesosoma, which resembles a polished stone.