'cough-suppressing' is a Modern English compound formed from 'cough' + the present participle 'suppressing' (from the verb 'suppress'). 'Cough' comes from Old English 'cohh' (also written 'coh'/'cohh'), and 'suppress' ultimately comes from Latin 'supprimere' (from 'sub-' meaning 'under' + 'primere'/'premere' meaning 'to press').

'suppress' entered English via Late Latin 'supprimere' and Old French 'supprimer'; its sense of 'pressing down' broadened to 'check or stop'. 'cough' has been used in Germanic languages (Old English 'cohh') for the respiratory sound. The compound 'cough-suppressing' is formed in modern English by combining the symptom noun and the participial form of the verb to describe something that suppresses that symptom.

Individually, 'suppress' originally meant 'to press down' but evolved to mean 'to check, restrain, or stop'. Combined with 'cough', the compound now specifically means 'that which reduces or stops coughing'.