may (v.1) Look up may at Dictionary.com
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 Look up May at Dictionary.com
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) Look up may at Dictionary.com
"to take part in May Day festivities," late 15c., from May. Related: Mayed; maying.