Apologies if this is "really" a geographical question, but I suspect it must have been addressed by scholars of the past millenium of Middle Eastern history, not least because it is so central to a long-running conflict (current as of early 2015):
Historically, Mesopotamia has been recognized as a coherent spatial concept. Its core is the land between the Tigris and Euphrates Rivers; its periphery has extended into and out of adjacent lands stretching from the Gulf in the southeast, the Zagros Mountains (and historic Persia) to the north and east, the Taurus Mountains (and historic Anatolia) to the far north, and a vast (and growing) desert to the west and south.
Syria has (IIUC) historically been rather more mutable, yet also remains conceptually distinct. Its most generally-agreed bounds are the Mediterranean to the west, the Taurus to the north, and those deserts to the east. Where one separates Syria from its southern neighbors, and what one calls the latter (e.g., Arabia, Egypt, the Levant, Palestine), seems to be endlessly contested.
But it seems pretty clear (ICBW) that, historically, populations and their attendant states have tended to settle in Mesopotamia or Syria, while the arid area "in between" has traditionally been a much less densely populated (and much poorer) borderland. So what has that mostly-desert area between Syria and Mesopotamia historically been called?
The only term I've found much used in English is the "Syrian desert," but mostly (like so many deserts) this area seems to be ignored by everyone except the people who live there ... at least until they attract the attention of outside powers.
