'antiguan' derives from the place name 'Antigua', which comes from Spanish 'Antigua' meaning 'ancient' (feminine form of 'antiguo').

'Antigua' (the island name) was borrowed into English from Spanish (named by Columbus as 'Santa María la Antigua'); the adjectival/demonym form 'antiguan' was formed in English from that place name.

Originally the Spanish word meant 'ancient'; as a proper noun it came to name the island, and 'antiguan' now primarily denotes origin or relation to that island.