'autorhythmic' is formed from the prefix 'auto-' (from Greek 'autos' meaning 'self') combined with 'rhythmic' (from 'rhythm' ultimately from Greek 'rhythmos' meaning 'measured flow').

'rhythm' entered English via Latin 'rhythmus' (from Greek 'rhythmos'); the compound 'autorhythmic' is a modern English formation (20th century) combining 'auto-' + 'rhythmic' to describe self-generated rhythmic activity in physiology.

Initially the components meant 'self' and 'rhythm' and the compound has kept that literal sense: 'producing rhythm by itself'; its use has become specialized in biology and medicine to describe self-paced cells or tissues.