Henry

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.

Proto-Indo-European root meaning "move in a straight line," with derivatives meaning "to direct in a straight line," thus "to lead, rule."

It forms all or part of: abrogate; address; adroit; Alaric; alert; anorectic; anorexia; arrogant; arrogate; bishopric; correct; corvee; derecho; derogate; derogatory; Dietrich; direct; dress; eldritch; erect; ergo; Eric; Frederick; Henry; incorrigible; interregnum; interrogate; maharajah; Maratha; prerogative; prorogue; rack (n.1) "frame with bars;" rail (n.1) "horizontal bar passing from one post or support to another;" Raj; rajah; rake (n.1) "toothed tool for drawing or scraping things together;" rake (n.2) "debauchee; idle, dissolute person;" rakish; rank (adj.) "corrupt, loathsome, foul;" real (n.) "small Spanish silver coin;" realm; reck; reckless; reckon; rectangle; rectify; rectilinear; rectitude; recto; recto-; rector; rectum; regal; regent; regicide; regime; regimen; regiment; region; regular; regulate; Regulus; Reich; reign; resurgent; rex; rich; right; Risorgimento; rogation; royal; rule; sord; source; subrogate; subrogation; surge; surrogate; viceroy.

It is the hypothetical source of/evidence for its existence is provided by:

Sanskrit raj- "a king, a leader," rjyati "he stretches himself," riag "torture" (by racking); Avestan razeyeiti "directs," raštva- "directed, arranged, straight;" Persian rahst "right, correct;" Latin regere "to rule, direct, lead, govern," rex (genitive regis) "king," rectus "right, correct;" Greek oregein "to reach, extend;" Old Irish ri, Gaelic righ "a king," Gaulish -rix "a king" (in personal names, such as Vircingetorix), Old Irish rigim "to stretch out;" Gothic reiks "a leader," raihts "straight, right;" Lithuanian raižytis "to stretch oneself;" Old English rice "kingdom," -ric "king," rice "rich, powerful," riht "correct;" Gothic raihts, Old High German recht, Old Swedish reht, Old Norse rettr "correct."