Etymology
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may (v.2)

"to celebrate May Day, to take part in May Day festivities," late 15c., from May. Maying as "the observance of May Day with all its sports and games" is attested from late 14c. (maiing).

And as a vapour, or a drop of raine
Once lost, can ne'r be found againe:
                     So when or you or I are made
                     A fable, song, or fleeting shade;
                     All love, all liking, all delight
                     Lies drown'd with us in endlesse night.
Then while time serves, and we are but decaying;
Come, my Corinna, come, let's goe a Maying.
[Robert Herrick, "Corinna's Going a-Maying," 1648] 
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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 root *magh- "to be able, have power." A present-preterit verb (with can, shall, etc.). Also used in Old English as a "auxiliary of prediction."

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day (n.)

Old English dæg "period during which the sun is above the horizon," also "lifetime, definite time of existence," from Proto-Germanic *dages- "day" (source also of Old Saxon, Middle Dutch, Dutch dag, Old Frisian di, dei, Old High German tag, German Tag, Old Norse dagr, Gothic dags), according to Watkins, from PIE root *agh- "a day."  He adds that the Germanic initial d- is "of obscure origin." But Boutkan says it is from PIE root *dhegh- "to burn" (see fever). Not considered to be related to Latin dies (which is from PIE root *dyeu- "to shine").

Meaning originally, in English, "the daylight hours;" it expanded to mean "the 24-hour period" in late Anglo-Saxon times. The day formerly began at sunset, hence Old English Wodnesniht was what we would call "Tuesday night." Names of the weekdays were not regularly capitalized in English until 17c.

From late 12c. as "a time period as distinguished from other time periods." Day-by-day "daily" is from late 14c.; all day "all the time" is from late 14c.  Day off "day away from work" is attested from 1883; day-tripper first recorded 1897. The days in nowadays, etc. is a relic of the Old English and Middle English use of the adverbial genitive.

All in a day's work "something unusual taken as routine" is by 1820. The nostalgic those were the days is attested by 1907. That'll be the day, expressing mild doubt following some boast or claim, is by 1941. To call it a day "stop working" is by 1919; earlier call it a half-day (1838). One of these days "at some day in the near future" is from late 15c. One of those days "a day of misfortune" is by 1936.

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maypole (n.)

also May-pole, "high striped pole decorated with flowers and ribbons for May Day merrymakers to dance around," attested from 1550s but certainly much older, as the first mention of it is in an ordinance banning them, and there are references to such erections, though not by this name, from a mid-14c. Welsh poem. See May Day.

It was usually cut and set up afresh on May-day morning, drawn by a long procession of oxen, decorated, as were also the pole itself and the wagon, with flowers and ribbons; but in some cases a pole once set up was left from year to year, as notably the famous pole of the parish of St. Andrew Undershaft in London, which was cut down in in reign of Edward VI. [Century Dictionary]
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may-fly (n.)

also mayfly, popular name of ephemeral insects of early spring, 1650s, from May + fly (n.).

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good-day (n.)
early 12c., "a fortunate day," also, generally, "good fortune;" from good (adj.) + day (n.). As a salutation in parting, haue godne day "have good day" is recorded from c. 1200; good day as a greeting is from late 14c.
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day-dream (n.)

also daydream, "a reverie, pleasant and visionary fancy indulged in when awake," 1680s, from day + dream (n.). As a verb, attested from 1820. Related: Day-dreamer; day-dreaming. Daymare "feeling resembling a nightmare experienced while awake" is from 1737.

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D-day (n.)

1918, "date set for the beginning of a military operation," with D as an abbreviation of day; compare H-hour, also from the same military order of Sept. 7, 1918:

The First Army will attack at H-Hour on D-Day with the object of forcing the evacuation of the St. Mihiel salient. [Field Order No. 8, First Army, A.E.F.]

"They designate the day and hour of the operation when the day and hour have not yet been determined, or where secrecy is essential" [U.S. Army Center of Military History Web site]. Now almost exclusively of June 6, 1944.

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field-day (n.)
1747, originally a day of military exercise and review (see field (v.)); figurative sense "any day of unusual bustle, exertion, or display" [Century Dictionary] is from 1827.
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latter-day (adj.)
"belonging to recent times," 1842; see latter (adj.). Originally in Latter-day Saints, the Mormon designation for themselves.
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