're-dye' is formed in modern English by combining the productive prefix 're-' (from Latin 're-' meaning 'again') with the verb 'dye' (English word meaning 'to color').

'dye' comes into English via Old and Middle English forms (and influences from Old French), and the Latin-derived prefix 're-' has long been used in English to indicate repetition; the combination 're-' + 'dye' produced 're-dye' in modern usage.

The root 'dye' originally meant 'to color' and has retained that basic sense; adding 're-' straightforwardly produced the meaning 'to color again', which has remained stable.