According to most sources, Buckeye grew up as the oilfield business increased. It was named for the Buckeye Sheep Ranch which was located in the area. At various times, it included a grocery store, as many as two gas stations, a cafe, welding shops, a hardware store and post office. (1)
In addition to the ranches, primarily the Lee, Eidson and Scharbauer ranches, there were a number of production camps including Phillips and Texaco, a large gasoline plant built by Phillips along with housing. The area also included a school at one time, and a Baptist church.
Below are some images of Buckeye as it probably looked in the 1950s and 1960s.
(1) Lynn C. Mauldin, Lea County New Mexico, A Pictorial History, The Donning Company, 1997.