homesickness n.

Old English ham "dwelling place, house, abode, fixed residence; estate; village; region, country," from Proto-Germanic *haimaz "home" (source also of Old Frisian hem "home, village," Old Norse heimr "residence, world," heima "home," Danish hjem, Middle Dutch heem, German heim "home," Gothic haims "village"), from PIE *(t)koimo-, suffixed form of root *tkei- "to settle, dwell, be home." As an adjective from 1550s. The old Germanic sense of "village" is preserved in place names and in hamlet.

'Home' in the full range and feeling of [Modern English] home is a conception that belongs distinctively to the word home and some of its Gmc. cognates and is not covered by any single word in most of the IE languages. [Buck]

Slang phrase make (oneself) at home "become comfortable in a place one does not live" dates from 1892 (at home "at one's ease" is from 1510s). To keep the home fires burning is a song title from 1914. To be nothing to write home about "unremarkable" is from 1907. Home movie is from 1919; home computer is from 1967. Home stretch (1841) is from horse racing (see stretch (n.)). Home economics as a school course first attested 1899; the phrase itself by 1879 (as "household management" is the original literal sense of economy, the phrase is etymologically redundant).

Home as the goal in a sport or game is from 1778. Home base in baseball attested by 1856; home plate by 1867. Home team in sports is from 1869; home field "grounds belonging to the local team" is from 1802 (the 1800 citation in OED 2nd ed. print is a date typo, as it refers to baseball in Spokane Falls). Home-field advantage attested from 1955.

1726, "morbid longing to return to one's home or native country, severe homesickness considered as a disease," Modern Latin, coined 1688 in a dissertation on the topic at the University of Basel by scholar Johannes Hofer (1669-1752) as a rendering of German heimweh "homesickness" (for which see home + woe).

From Greek algos "pain, grief, distress" (see -algia) + nostos "homecoming," from neomai "to reach some place, escape, return, get home," from PIE *nes- "to return safely home" (cognate with Old Norse nest "food for a journey," Sanskrit nasate "approaches, joins," German genesen "to recover," Gothic ganisan "to heal," Old English genesen "to recover"). French nostalgie is in French army medical manuals by 1754.

Originally in reference to the Swiss and said to be peculiar to them and often fatal, whether by its own action or in combination with wounds or disease.

[Dr. Scheuzer] had said that the air enclosed in the bodies of his countrymen, being in Æquilibrium with a rare and light air that surrounds them, was overloaded in lower countries with an air more dense and heavier, which compressing and obstructing the capillary vessels, makes the circulation slow and difficult, and occasions many sad symptoms. [Account of the publication of "Areographia Helvetiæ" in New Memoirs of Literature, London, March 1726] 

By 1830s the word was used of any intense homesickness: that of sailors, convicts, African slaves. "The bagpipes produced the same effects sometimes in the Scotch regiments while serving abroad" [Penny Magazine," Nov. 14, 1840]. It is listed among the "endemic diseases" in the "Cyclopaedia of Practical Medicine" [London, 1833, edited by three M.D.s], which defines it as "The concourse of depressing symptoms which sometimes arise in persons who are absent from their native country, when they are seized with a longing desire of returning to their home and friends and the scenes their youth ...."

It was a military medical diagnosis principally, and was considered a serious medical problem by the North in the American Civil War:

In the first two years of the war, there were reported 2588 cases of nostalgia, and 13 deaths from this cause. These numbers scarcely express the real extent to which nostalgia influenced the sickness and mortality of the army. To the depressing influence of home-sickness must be attributed the fatal result in many cases which might otherwise have terminated favorably. ["Sanitary Memoirs of the War," U.S. Sanitary Commission, N.Y.: 1867]

Transferred sense (the main modern one) of "wistful yearning for the past" is recorded by 1920, perhaps from such use of nostalgie in French literature. The longing for a distant place also necessarily involves a separation in time.