In the ancient times it was called,, Aramnahrin" (or: Aram-Nahrin) in Aramaic. In Hebrew it was called,, Aram-Naharaim" which we encounter in the old testament. When the old testament was translated into Greek, the term Aram-Naharaim was translated by Mesopotamia; which many get familiarized with during their education.
Because of overwhelming presence of Arameans in this part of the world, in the era of early Christianity, it was called Beth-Aramaya, meaning (in Aramaic) the house of the Arameans. The northern part of Aram-Naharaim was called Paddan-Aram, meaning the land of Aram where Biblical figures like Abraham and Jacob have lived and walked, which is roughly speaking situated in southeastern Turkey. A part of Paddan-Aram is today called Tur Abdin in the Aramaic (Syriac) language, which means “the mountains of the servants of the Lord”, because of the overwhelming presence of monasteries and churches.
West- Arameans, East- Arameans
The Aramean nation was at beginning of early Christianity geographically divided into East- Arameans and West- Arameans.
The river the Euphrates in general served as the border. Thus those living east of Euphrates, roughly speaking Iran, were called East- Arameans. And those living at the West of Euphrates were called West- Arameans. The East- Arameans, also called Nestorians, in general were volatile and as has been explained numerous groups emanated from them.
More about this: http://www.aramnahrin.org/English/iraqichristians.htm
Beth- Aramaye or Beth- Ormoye
Beth- Aramye: Eastern (Aramaic) Syriac dialect
Beth- Oromoye: Western (Aramaic) Syriac dialect.
This term is used by quite number of famouse scholars of both, the West- Syrian as well as the east- Syrian Church
East- Syrian Church: Simon Bar Sabbae (2nd Catholicos of Seleucia- Ktiphon); Catholicos- Patriarch Elias (I) of Tirhan (= Terehan) (1028-1049); Mor Touma Audo, born in Alqosh [Iraq] in 1855 and killed in the year of 1917; Ya‘qob Avgin (= Eugene) Manna (1867-1928) born in Bet Qopa [Iraq];
West- Syrian Church: Saint James of Sarug (Sürüç, Turkey, †512); Simeon of Beth Arsham; The Anonymous of Zuqnin, Syrian-Orthodox (West-Aramean) Patriarch St. Michael the Great of Militene;
1 Chronicle 19:6 “ King Hanun and the Ammonites realized that they had made David their enemy, so they paid nearly forty tons of silver to hire chariots and charioteers from Upper Mesopotamia (= Aram-Naharaim) and from the Aramean states of Maacah and Zobah.
Deuteronomy 23:4 “ They refused to provide you with food and water when you were on your way out of Egypt, and they hired Balaam son of Beor, from the city of Pethor in Mesopotamia (= Aram-Naharaim), to curse you. .
Genesis 24:10,, The servant, who was in charge of Abraham's property, took ten of his master's camels and went to the city where Nahor had lived in northern Mesopotamia (= Aram-Naharaim).
Acts 2:9,, We are from Parthia, Media, and Elam; from Mesopotamia (=Aram-Naharaim), Judea, and Cappadocia; from Pontus and Asia, ,
Psalm 60:2,, For the choir director; according to shushan eduth; a miktam by David; for teaching. When David fought Mesopotamia (= Aram Naharaim) and Aram Zobah, and when Joab came back and killed 12,000 men from Edom in the Dead Sea region. O God, you have rejected us. You have broken down our defenses. You have been angry. Restore us
Judges 3:8,, So the LORD became angry with Israel and let King Cushan Rishathaim of Mesopotamia (= Aram-naharaim) conquer them. They were subject to him for eight years. .