Give a damn is a common expression to convey the level of concern or interest in a particular matter. There are several synonyms that can be used to replace this phrase, depending on the context and the mood of the speaker. For instance, care, value, and worry are all synonyms that convey a sense of importance or significance to something. Other alternatives to give a damn include showing interest, attentiveness, concern, or consideration. Expressions such as being invested, taking notice, paying attention, or giving heed are also synonyms that can be used to replace give a damn. Ultimately, the choice of synonym will depend on the situation and the speaker's personal preference.