Hauyne forms a solid solution series with Lazurite, essentially sharing the same chemical formula, but with a variation of sulfide over sulfate. Hauyne has the sulfate radical dominating, whereas Lazurite has the sulfide element dominating. Much of the known Lazurite specimens are actually sulfate-dominating, which means they are in fact Hauyne and not Lazurite. This is especially true at the most famous locality for Lazurite at Sar-e-Sang in the Kokcha Valley of Afghanistan, where the material labelled as Lazurite has been determined to really be Hauyne. Despite this, the mineral community still generally accepts this material as Lazurite notwithstanding the scientific inaccuracy. It is generally accepted to refer to the opaque, non-fluorescent, ultramarine-blue material as Lazurite, whereas Hauyne describes the transparent to translucent material.

Hauyne was named in 1807 in honor of French mineralogist and crystallographer Abbé Rene Just Haüy (1743-1822), curator of the Muséum National d'Histoire Naturelle in Paris.