Both Josephus and the Septuagint translate the name as Mesopotamia. Ancient writers elsewhere used the name "Mesopotamia" for the land between the Tigris and Euphrates. According to the Book of Jubilees, when the entire Earth was divided among the sixteen grandsons of Noah, Aram, the son of Shem received as an inheritance for his offspring, lands bordered by the Euphrates and the Tigris (Jubilees 9:5); it also associates the city of Ur Kesed not with the descendants of Aram, but rather with those of Arphaxad, his brother, who was Abram's ancestor.
However the usage of the Hebrew name "Aram-Naharaim" does not match the general usage of "Mesopotamia", the former being used exclusively for a northern region. Moreover the translation of the name as "Mesopotamia" was not consistent - the Septuagint also uses a more precise translation "Mesopotamia of Syria" as well as "Rivers of Syria". Josephus refers to the subjects of Chushan, king of Aram Naharaim, as "Assyrians".
Hebrew has a distinct name Ashur for the region of Assyria containing the Tigris. Aram Naharaim lay west of Ashur as it contained Haran. Haran itself lies on the west bank of the Balikh, east of the Upper Euphrates. The traditional Jewish location of Ur Kasdim (at Edessa) and the Balikh itself lie west of the Khabur implying that the second river was understood to be the latter by those maintaining this tradition.