Definition of day-day in English:
day-day
exclamation ˈdeɪˌdeɪˌdeɪˈdeɪ
informal ‘Goodbye’. Also as noun in "to shake a (also one's) day-day" and variants: to wave goodbye.
Origin
Late 17th century; earliest use found in Elkanah Settle (1648–1724), playwright. Reduplication of day.