may (v.1)
Old English mæg "am able" (infinitive magan, past tense meahte, mihte), from Proto-Germanic root *mag-, infinitive *maganan (Old Frisian mei/muga/machte "have power, may;" Old Saxon mag/mugan/mahte; Middle Dutch mach/moghen/mohte; Dutch mag/mogen/mocht; Old High German mag/magan/mahta; German mag/mögen/mochte; Old Norse ma/mega/matte; Gothic mag/magan/mahte "to be able"), from PIE *magh- (1) "to be able, have power" (cognates: Greek mekhos, makhos "means, instrument," Old Church Slavonic mogo "to be able," mosti "power, force," Sanskrit mahan "great"). Also used in Old English as a "auxiliary of prediction."
May
fifth month, early 12c., from Old French mai and directly from Latin Majus, Maius mensis "month of May," possibly from Maja, Maia, a Roman earth goddess (wife of Vulcan) whose name is of unknown origin; possibly from PIE *mag-ya "she who is great," fem. suffixed form of root *meg- "great" (cognate with Latin magnus). Replaced Old English þrimilce, month in which cows can be milked three times a day. May marriages have been considered unlucky at least since Ovid's day. May-apple attested from 1733, American English.
may (v.2)
"to take part in May Day festivities," late 15c., from May. Related: Mayed; maying.