guest-room (n.)

Old English gæst, giest (Anglian gest) "an accidental guest, a chance comer, a stranger," from Proto-Germanic *gastiz (source also of Old Frisian jest, Dutch gast, German Gast, Gothic gasts "guest," originally "stranger"), from PIE root *ghos-ti- "stranger, guest, host" (source also of Latin hostis, in earlier use "a stranger," in classical use "an enemy"); the root sense, according to Watkins, probably is "someone with whom one has reciprocal duties of hospitality."

Spelling evolution influenced by Old Norse cognate gestr (the usual sound changes from the Old English word would have yielded Modern English *yest). Meaning "person entertained for pay" (at an inn, etc.) is from late 13c. Old English also had cuma "stranger, guest," literally "a comer." Phrase be my guest in the sense of "go right ahead" first recorded 1955.

Middle English roum, from Old English rum "space, extent; sufficient space, fit occasion (to do something)," from Proto-Germanic *ruman (source also of Old Norse, Old Saxon, Old High German, Gothic rum, German Raum "space," Dutch ruim "hold of a ship, nave"), nouns formed from Germanic adjective *ruma- "roomy, spacious," from PIE root *reue- (1) "to open; space" (source also of Avestan ravah- "space," Latin rus "open country," Old Irish roi, roe "plain field," Old Church Slavonic ravinu "level," Russian ravnina "a plain").

Old English also had a frequent adjective rum "roomy, wide, long, spacious," also an adverb, rumlice "bigly, corpulently" (Middle English roumli).

The meaning "chamber, cabin" is recorded by early 14c. as a nautical term; applied by mid-15c. to interior division of a building separated by walls or partitions; the Old English word for this was cofa, ancestor of cove. The sense of "persons assembled in a room" is by 1712.

Make room "open a passage, make way" is from mid-15c.  Room-service is attested from 1913; room-temperature, comfortable for the occupants of a room, is so called from 1879. Roomth "sufficient space" (1530s, with -th (2)) now is obsolete.