'arch-idiot' is formed from the prefix 'arch-' and the noun 'idiot'. 'arch-' ultimately comes from Greek 'arkhē/arkhos' meaning 'chief, principal', used in English as an intensifying prefix. 'idiot' originates from Greek 'idiōtēs', meaning 'private person; layman', later 'person lacking skill or intelligence'.

'arch-' entered English via Latin/Old French as a combining form ('arch-'/'archi-') and has been used since Middle English in compounds like 'arch-enemy'. 'idiōtēs' passed into Latin as 'idiota', into Old French and then Middle English as 'idiot', developing the modern English form 'idiot.' The compound 'arch-idiot' is a modern formation combining the two elements.

Originally, 'idiōtēs' meant 'private person' (someone not holding public office or specialized knowledge). Over time it shifted to mean 'uninformed person' and then 'fool' or 'stupid person', while 'arch-' developed as an intensifier meaning 'principal' or 'extreme', so 'arch-idiot' now means 'an extreme/fundamental idiot.'