As in the title, my question is whether the Romans had a name with which they referred to the Sahara itself as a geographical location, in addition to their names for African regions such as "Aethiopia" or "Mauretania."
There were a few different terms for it, but it didn’t really have a name. I guess we tend to think of the word “deserted” as referring to a desert, but it’s actually the other way around. The word “desert” comes from the Latin verb desero, which means to abandon or give up something, or desert your post in a military sense. The past participle of the verb, desertus, meant a place that was abandoned or a wasteland. When used as a noun, it could be neuter singular (desertum) or plural (deserta).
Most of the time there doesn’t seem to be a particular name for it other than “the desert”, the vast desert (deserta vasta) or the great desert (desertum magnum). The Romans probably wouldn’t have understood those as names though, just a description of what they considered to be a big wasteland. Sometimes they also apparently considered it a kind of sea, like the counterpart to the mare magnum, the Mediterranean. In this sense they also called it the sandy sea, mare arenosum.
Parts of had a name if they were associated with the people who lived nearby. The part to the west of Egypt was the Libyae desertum, the desert of Libya, which is not really modern Libya but more like eastern Libya and western Egypt. But that name could also apply by extension to the whole thing. Virgil uses this name in book I of the Aeneid (line 384). Otherwise it could also be called the Africae desertum, the desert of Africa, which is what Pliny calls sometimes. This could also refer to the whole desert, but strictly speaking, Roman Africa means parts of modern western Libya, Tunisia, and eastern Algeria.
In fact it doesn’t even really have a modern name either! The “Sahara Desert” literally means “Deserts Desert”. Sahara is the Arabic plural of a word that simply means a deserted wasteland (like Latin deserta). In Arabic it is sometimes also called “Sahara kubra”, the great desert(s).
Sources:
Katia Schörle, "Saharan trade in classical antiquity", in Saharan Frontiers: Space and Mobility in Northwest Africa (Indiana University Press, 2012)
Jean Desanges, Toujours Afrique apporte fait nouveau (De Boccard, 1999)
Brian Warmington, The North African Provinces (Cambridge University Press, 1954)
Susan Raven, Rome in Africa, 3rd ed. (Routledge, 1993)