'archrival' is formed from the prefix 'arch-' (from Greek 'arkhē'/'arkhós' meaning 'chief, principal') combined with 'rival' (from Latin 'rivalis', originally 'one using the same stream', meaning 'one who competes').

'rival' came into English via Latin 'rivalis' and Old French 'rival'; the combining prefix 'arch-' (from Greek) was attached in Modern English to create 'archrival' to indicate the 'chief' rival, giving the compound 'archrival'.

Initially the elements meant 'chief' + 'one who competes (or shares a stream)'; over time the compound came to be used for a person's or group's principal opponent and has retained that core sense.