There were two periods in the Assyrian and Akkadian/Babylonian histories: the early and late, divided by nearly a 1000 years. The early periods coincide with the period of the decline of the Sumerians. In fact the decline of the Sumerians may have been caused by the arrival of these Semites from the south and the west (the Amurites/Aramian) that got worse by the arrival of the Hurrians from the north (the Gutils), and even the Dravidians (the Elamites) from the east. The poor things were squeezed out of everything.
We know of the Hurrian Haza who lived west of the Lake Van, then gradually moved to where Arbil is. By the time the Medians took Assyria, that entire area had been Kurdified by the formerly Hurrian Haza and then the Aryan Sagartha. Since we have NO written record of any kind from the Medians (they may have been totally illiterate people as the Persian were until the time of Cyrus). However, the occurrence of the name Hadeap/Haziab/Haziav is from this period. What is interesting is that linguistically, the name must have been at least several centuries old because while "Haza" is Hurrian, the rest of the name is Aryan. It must have gone this way: Hazawand, Hazaband/Hadhaban, Hazaban/Hadhaban, Hadhiab, from whence the Greek mispronunciation of Adeap and Adiabene.
The Sumerian "Ur Billum" was an attempt by them to make sense of the Semitic name of the place for which they had no understanding. Arba Ilu, "four gods", is a clear reference to the city of the four gods of Amurite/Assyrian/Akkadian. These were Shamash (sun), Sin (moon), Ishtar (Venus), Neregal (Mars)--headed by the supreme deity, Assur (for the Assyrians) or Marodukh (for the Babylonians). The Sumerians tried to read the name as Ur Billum, meaning The City of Billum, which means really nothing. The name is NOT Sumerian, but Semitic.
The Median name of Hadeap was the direct reference to the Hadhabani (Hadha-wand/Hazawand) ancestors of Kurds who had settled there. The place and the city came to be known as Adiabene to the classical world, named after these Kurds. King Saladin is known from his contemporary biographies as being a Hadhabani/Hazabani Kurd.
Therefore, the best choice for the city's name is not the two mispronunced names of Hewler or Irbil/Erbil. The Kurdish name should Hazaban/Hadhban (whence, Adiabene)
There is no need to prove Hewler is just a terribly distorted Kurdish mispronunciation of the old name, Arbil/Arbela, because it plainly is!
http://www.kurdishacademy.org/?q=node/673
