'cough-suppressant' is a modern English compound formed from 'cough' + 'suppressant'. 'cough' originates from Old English, specifically the word 'cohhian' (an imitative/onomatopoeic word for the sound of coughing), while 'suppressant' ultimately originates from Latin via French, from the Latin root 'supprimere' where 'sub-' meant 'under' and 'premere' meant 'to press'.

'suppressant' developed from Latin 'supprimere' > Old French 'supprimer' > Middle English 'suppress' with the agentive suffix '-ant' producing 'suppressant' in later medical/technical usage. 'cough' comes from Old English 'cohhian' and remained an imitative word into Modern English; combining the two produced the compound 'cough-suppressant' in modern usage.

Originally 'supprimere' meant 'to press under' and 'suppress' meant 'to press down'; over time the sense shifted toward 'to stop or restrain', so 'suppressant' came to mean 'an agent that stops or reduces' (e.g., coughing). 'cough' has retained its onomatopoeic origin while combining with 'suppressant' gave a medical sense.