Old English dæge "female servant, woman who handles food in a household, housekeeper," from Proto-Germanic *daigjon (source also of Old Norse deigja "maid, female servant," Swedish deja "dairymaid"), from PIE root *dheigh- "to form, build." 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.
OED says the ground sense of the ancient word seems to be "kneader, maker of bread;" it would have then advanced via Old Norse deigja and Middle English daie to mean "female servant, woman employed in a house or on a farm." By c. 1200 it had acquired the specific sense of "woman in charge of milking and making butter and cheese, dairy-maid." Dæge as "servant" is the second element in many surnames ending in -day (such as Faraday, and perhaps Doubleday, if it means "servant of the Twin," etc.).
title of a military commander in Muslim north Africa, 1650s, from Turkish dai "maternal uncle," a friendly title used of older men, especially by the Janissaries of Algiers of their commanding officers. As these often became rulers in the colony it was used in English as the title of governor of Algiers under Ottoman rule, There were also deys in Tunis and Tripoli.