'agate-decorated' is a compound of 'agate' and 'decorate'. 'agate' originates from Greek, specifically the word 'ἄχατης (akhátēs)', where the name referred to the river Achates in Sicily and came to mean the banded chalcedony gemstone; 'decorate' originates from Latin, specifically the word 'decorare', where 'decor' meant 'beauty, properness'.

'agate' passed from Greek 'akhátēs' into Latin as 'achates', then into Old French and Middle English as 'agate'. 'decorate' comes from Latin 'decorare', passed into Old French as 'decorer' and into Middle English as 'decorate'; the compound form 'agate-decorated' is a modern English compound built from these elements.

Initially, 'agate' referred specifically to the gemstone named after a location and 'decorate' meant to adorn; together the compound's meaning is straightforwardly 'adorned with agate' and has not undergone significant semantic shift beyond compositional combination.