Aram Naharaim is a region mentioned in the Tanach. It is the region where Abraham was born (Gen 24:2-10  ) and had lived many years (Gen 11:31-12:4 ), from where the mothers of the Jewish people, Sarah, Rebecca, Rachel and Leah had come (Gen 11:29-31, 24:10-15, 28:2, 29:16  ), where Jacob had lived after fleeing from Esau (Gen 28:5 ), where all his sons except Benjamin had been born (Gen 29:32-30:26, 35:16-18 ).
The word Naharaim can be understood to mean "two rivers" - the dual of Nahar meaning "river". Thus a common assumption is that this refers to the two major rivers in the Middle East - the Tigris and Euphrates and hence that Aram Naharaim is equivalent to Mesopotamia - Greek for "land between the rivers" - referring to the valley between the Tigris and Euphrates. The Septuagint thus translates Aram Naharaim as "Mesopotamia" on occasion  as does Josephus [4a].
However the places traditionally associated with Aram Naharaim, such as the cities of Haran, Nahor and Ur Kasdim (traditionally located at Edessa  ) all lie between the headwaters of the Euphrates in a region that can at most be considered the northwest corner of "Upper Mesopotamia" - they do not lie within the valley between the Tigris and the Euphrates proper which is what is usually meant by Mesopotamia. Moreover, Babylon the most famous Mesopotamian city is never referred to as being in Aram Naharaim. Indeed, the Septuagint sometimes translates Aram Naharaim as "Mesopotamia of Syria" or as "Rivers of Syria"  and Josephus refers to the people of Aram Naharaim as "Assyrians" [4b], indicating that it was not the whole of Mesopotamia but a northern region. The Tanach however distinguishes between the region of Asshur (Assyria, of which "Syria" is a shortened form [7a]) and Aram Naharaim which clearly lay to the west of the former since it contained Haran.
The El-Amarna letters mention a place called Naharima in Akkadian, referring to the region of the Upper Euphrates and its tributaries  [9a]. This name appears to be the equivalent of Naharaim  [9a]. The Egyptians referred to the region as Naharin, which was also the name for the Upper Euphrates  [9a]. Naharaim can also be understood to mean "along the river" or "within the river" . Thus all the evidence points to Aram Naharaim being the region lying between the headwaters of the Euphrates proper - the land bounded by the Habor (the easternmost headwater of the Euphrates proper) and the Naharin (or Upper Euphrates - the westernmost headwater of the Euphrates proper).
The Tanach tells us that the eastern border of Biblical Israel is the Euphrates (Gen 15:18, Deu 11:24 ). The Tanach also tells us that Israel includes the land of the Kadmonites (Gen 15:19 ). The Kadmonites (Qadmonim) or "Easterners" were Abraham's family living in the east - initially his relatives in Aram Naharaim (Gen 28:5-29.1  [10a]), later his descendents living east of the Israelites. They are also called Bnei Qedem and Genesis 29:1 explicitly refers to the region in which Haran lay as being part of the land of the B'nei Qedem. This shows that the land lying between the headwaters of the Euphrates - Aram Naharaim - was part of Biblical Israel, the precise eastern border in the region of the headwaters being the easternmost headwater, not the westernmost.
Tanach,.Masoretic Text,. Mechon Mamre, Jerusalem, 2002
RaMBaN on Lech Lecha, Gates of Jewish Heritage, 2002-2003 (Here it is explicitly pointed out that Abraham's birthplace, Ur Kasdim, was in Aram Naharaim.)
Septuagint, Greek text, Michael Hagget, 2002-2003 (The Septuagint translations of Gen 24:10 and Deu 23:4 refer to Aram Naharaim as "Mesopotamia", but the translation of Jdg 3:8 refers to it as "Rivers of Syria" while the translations of 1 Ch 19:6 and Psa 60:1 call it "Mesopotamia of Syria")
Flavius Josephus, Antiquities of the Jews, in Flavii Iosephi opera, ed. B. Niese, Weidmann, Berlin, 1892
Book 1, Chapter 6, 5 (Here Josephus refers to Haran as being in Mesopotamia using this name for the region called Aram Naharaim in Gen 24:10.)
Book 5, Chapter 3, 2 (Here Josephus refers to Cushan-rishathaim, the king of Aram Naharaim in Jdg 3:8, as king of the Assyrians.)
Abraham, Theophilus G. Pinches, in The Old Testament In the Light of the Historical Records and Legends of Assyria and Babylonia, Society for Promoting Christian Knowledge, London, 1903, revised edition 1908
Of Syria and Mesopotamia in A Description of the East, Richard Pococke, London, 1743, vol. II, p. 88 - 209 (Here, in the 18th century, Pococke notes a universal agreement amongst Jews that Edessa was Ur Kasdim.)
Herodotus, The Histories, in Herodotus, with an English translation ed. A. D. Godley, Harvard University Press, Cambridge, 1920
Book 7, 63 (Here Herodotus explains that "Syrians" is the Greek name for the Assyrians revealing that the name Syria arose as a Greek shortened form of Assyria.)
Shabbat Hayei Sarah in Margoliot Shel Torah - Pearls of Torah, Rabbi David M. Greenspoon, 1999 (Note that here Naharin is transliterated as Naharain. The ancient Egyptians did not write vowels hence the exact vowelization is unknown.)
The Jewish Encyclopedia, 1901-1906, online edition JewishEncyclopedia.com, 2002
article Aram-Naharaim (Note that here Naharima is transliterated as Nahrima, Akkadian vowels being uncertain due to ambiguities in their writing system.)
Encyclopedia Judaica - CD-ROM Edition Version 1.0, Judaica Multimedia, Jerusalem, 1997
article Kadmonites (Here it is explicitly pointed out that the Kadmonites and the B'nei Qedem are the same people. That the Kadmonites included Abraham's relatives in the region of Haran is seen from Gen 28:5-29.1.)