English, French, Danish, Dutch, and German: from a short form of the personal name Matthias (see Matthew) or any of its many cognates, for example Norman French Maheu.
English, French, Dutch, and German: from a nickname or personal name taken from the month of May (Middle English, Old French mai, Middle High German meie, from Latin Maius (mensis), from Maia, a minor Roman goddess of fertility). This name was sometimes bestowed on someone born or baptized in the month of May; it was also used to refer to someone of a sunny disposition, or who had some anecdotal connection with the month of May, such as owing a feudal obligation then.
English: nickname from Middle English may "young man or woman".
Irish (Connacht and Midlands): when not of English origin (see 1 - 3 above), this is an Anglicized form of Gaelic Ó Miadhaigh "descendant of Miadhach", a personal name or byname meaning "honorable", "proud".
French: habitational name from any of various places called May or Le May.