Old English dæge "female servant, housekeeper, maid," from Proto-Germanic *daigjon (cognates: Old Norse deigja "maid, female servant," Swedish deja "dairymaid"), from PIE *dheigh- "to form, build" (see dough). Now obsolete (though OED says, "Still in living use in parts of Scotland"), it forms the first element of dairy and the second of lady.
The ground sense seems to be "kneader, maker of bread;" advancing by Old Norse deigja and Middle English daie to mean "female servant, woman employed in a house or on a farm." Dæge as "servant" is the second element in many surnames ending in -day (such as Faraday, and perhaps Doubleday "servant of the Twin," etc.).
1650s, "title of a military commander in Muslim north Africa," from Turkish dai "maternal uncle," a friendly title used of older men, especially by the Janissaries of Algiers of their commanding officers. There were also deys in Tunis and Tripoli.
