With indefinite article and modifying adjective: a reputation of a specified kind. Now frequently in to give (also get) a good (also bad) name.

The reputation of a specified person, place, institution, etc.; spec. = reputation, n. 4. Also: a particular reputation gained by a person. Now rare.

Estimation or esteem of a specified kind or degree.

U.S. Prison slang and Criminals' slang. A reputation; esp. one which is bad or unfavourable. Often with preceding modifying word specifying the…

= credibility, n. 1a; reputation or status among one's peers.