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.

"inanimate objects that, when contaminated with or exposed to infectious agents, can retain and transfer the disease," plural of fomes (1650s), which is from medical Latin fomes (used in this sense first by Fracastoro, 16c., probably on the notion of "fuel"), from Latin fomes, fomitis "kindling-wood, touchwood, tinder," from fovere "to warm, keep warm" (see fever). The classically incorrect back-formed singular fomite is attested from 1859.

also hay-fever, 1825, from hay + fever. Also called summer catarrh (1828); not much noted before the 1820s, when it was sometimes derided as a "fashion" in disease.

People are apt to sneeze, in hot weather for example; and people do not die of sneezing now-a-days, as they did in days that no one knows any thing about. We cannot give six draughts a-day, at one and nine pence each, for sneezing: call it the hay-fever. What a wonderful man! what a clever man! he understands the hay-fever: call him in! Thus is the hay-fever among the last in the list of fashionables. ["On Fashions in Physic," London Magazine, October 1825]