It seems a lot of groups are given names after the fact (I'm thinking of posts I've seen here about the the Hittites and possibly the Cathars). What is the oldest society that we know how they identified themselves, and (if it's not the same) what is the oldest society that we still commonly use their own identifier to refer to them? Also, why did they chose these names for themselves?
EDIT: Also, I am called an American because I come from the land that was named after the guy who determined it was its own continent and not part of Asia. Romans were presumably named after the city that they came from. Is this pretty standard when cultures decide names for themselves? What other examples are there of people naming themselves after things other than where they dwell?
The "Sumerian"(in scare quotes because defining what that meant in c. 2300 BCE is borderline impossible to pin down) self-identified as the "Black-headed people" in our extant Sumerian literary texts, but it's still very uncertain why they called themselves the "black-headed people".
As for the oldest name for a culture or society that is still in use today, Iran is one of several good candidates. It is attested as Ērān and Ērānšhar in the inscriptions of Ardashir I, founder of the Sasanian dynasty(180-242 CE) but is related linguistically to the much older Avestan term Airyanəm Vaējah(roughly, Land of the Aryans) and the reconstructed Old Persian *aryānām xšaθra; although Iran is not attested as a geographical term in the Old Persian inscriptions, the closely related Aryan is attested as a ethnoliguistic designator. Thus for example at Bisotun Darius I refers to himself both as a Persian and "An Aryan, having Aryan lineage" and according to the Encyclopedia Iranica the Elamite text of the Bisotun inscription glosses Ahura Mazda as "god of the aryas".