birthday (n.)

c. 1200, "fact of being born;" mid-13c., "act of giving birth, a bringing forth by the mother, childbirth," sometimes in Middle English also "conception;" also "that which is born, offspring, child;" from a Scandinavian source such as Old Norse *byrðr (replacing cognate Old English gebyrd "birth, descent, race; offspring; nature; fate"), from Proto-Germanic *gaburthis (source also of Old Frisian berd, Old Saxon giburd, Dutch geboorte, Old High German giburt, German geburt, Gothic gabaurþs), from PIE *bhrto past participle of root *bher- (1) "to carry; to bear children" (compare bear (v.)).

The Germanic suffix -th is for "process" (as in bath, death). The meaning "condition into which a person is born, lineage, descent" is attested from c. 1200 (it was also in the Old English word). In reference to non-living things, "any coming into existence" is from 1610s. Birth control is from 1914; birth certificate is from 1842.

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.