'well-dyed' is a compound of the adverb 'well' and the past participle 'dyed'. 'well' originates from Old English 'wel', meaning 'thoroughly' or 'well', and 'dye' comes from verbs in Old English/Old French meaning 'to color'.

'well' remained from Old English into Modern English with its meaning of 'thoroughly'; 'dyed' is the past-participle form of 'dye', which appears in Middle English as forms derived from Old English/Old French roots for 'color'. The compound adjective 'well-dyed' developed in Modern English by combining these elements.

Initially the elements meant 'thoroughly' + 'colored'; over time the compound kept that core sense and is used to describe fabric (or figuratively appearance) that is thoroughly or richly colored.