where is eunice nm
- Eunice, New Mexico. Eunice is a city in Lea County, New Mexico, United States. The population was 2,562 at the 2000 census. The mayor of Eunice, as of April 2007, is Matt White. Ground was broken for construction of a National Enrichment Facility, which uses Zippe-type centrifuge technology, to enrich uranium in August, 2006.