'sound-minded' is a Modern English compound formed from the adjective 'sound' (meaning 'healthy, whole') + the participial/adjectival element 'minded' (from 'mind').

'sound' (meaning 'healthy, whole') comes from Old English forms such as 'gesund'/'sund' (cognate with German 'gesund'), ultimately from Proto-Germanic roots meaning 'healthy'; 'mind' comes from Old English 'gemynd' meaning 'memory, thought', and the compound 'sound-minded' developed in post‑Medieval English by combining these elements to describe a person of a healthy or well-ordered mind.

Initially it conveyed the literal sense of 'having a healthy mind', and over time it has retained that meaning while also being used to describe someone 'having sound judgment' or 'sensible'.