"To coin a phrase" is a common expression used when introducing a new term or when a speaker wishes to emphasize a particular phrase. However, if you're looking for alternative phrases or synonyms to describe this action, then there are several options available to you. For example, you could say "to create a term," "to invent a saying," "to come up with a new expression," or "to introduce a buzzword." Other synonyms for "to coin a phrase" include "to formulate a catchphrase," "to originate a slogan," "to devise a byword," or "to devise a watchword." Each of these phrases offers unique ways to describe inventing new phrases that are memorable and catchy.