The first mentions of the Kurds as a defined group in the historical record begin in the 7th century. There are references to Kurds in both Arab and Sassanid documents. However, this was probably not an "ethnic" definition as people of that time did not conceive of ethnicity the way we do today. The term could may have been used in reference to any of the semi-nomadic tribes speaking an Iranian language.

The Shahnameh gives a very interesting myth of their origin, claiming they are descended from people saved from the evil Arab king Zahak who had to feed two human brains a day to the snakes growing out of his shoulders. The Shahnameh was written in the late 10th century, but based on pre-Islamic mythology and traditions.

You should not trust Kurdish propaganda claimimg they are descendants of the Medians. Almost all modern historians reject that theory. Turkish propaganda claiming they are "mountain Turks" is also incorrect.