A mourning veil was a black veil worn on the head while at a funeral.
In 2367, "Hercule-Savinien De Cyrano de Bergerac" (Reginald Barclay) asked "Roxane" (Beverly Crusher) that she "allow a double meaning for your mourning veil." (i.e. mourn both him and Christien de Neuvillette). (TNG: "The Nth Degree")