Origin of "hang tight"?
"Hang tight" goes back to at least 1901:
Hang tight, my friends ! Hang tight ! Hang tight ! " said he, when, suddenly, one near the top, in the agitation of the moment, began to sneeze, lost his hold, and down the whole string, hundreds of them, fell, and were completely flattened out
Or is it 1879?
Hang tight. Good —. What I Thank God! Thank God I We are saved ! Saved ! Saved ! Bertha we are saved !
Or maybe 1854?
“Hold on, brave spars; hang tight, good rope; a whole life now rests on every strand.”
Clearly it refers to holding on, either actually or metaphorically.
By 1922 the metaphorical meaning was becoming more common:
The mere fact that prices of raw furs are advancing is the best indication of a business revival. It is a well-known fact that the buying public hang tight to their cash on a falling market.
And that sense more or less carried through to 2016:
And so they were going to wait until they're retired or semi-retired in their cubical-land day job that they don't enjoy. They're going to hang tight. Then the recession happened and they sure as hell were going to hang tight.” Kurth has changed his business model and renamed it PivotPlanet. After the 2008 recession, business dropped off precipitously.
According to the Phrase Finder it may be a boxing term probably in use from the 60s:
Hang tight:
Hang to the rigging, be patient. 1966. "Dictionary of American Regional English," Volume II by Frederic G. Cassidy and Joan Houston Hall (1991, Belknap Press of Harvard University Press, Cambridge, Mass., and London, England). Page 893.
Hang in there -- Stick with it, even though the going is tough. Probably a boxing term for a fighter "momentarily getting the worst of it" who clings "to the ropes or the arms of his opponent for a respite." Used in a 1972 The Atlantic article about President Nixon: "...it would be in his nature to hang in there and fight." "Dictionary of Cliches" by James Rogers (Wings Books, Originally New York: Facts on File Publications, 1985). Page 120.
According to The Concise New Partridge Dictionary of Slang and Unconventional English the expression is an AmE one from 1947.
My impression is that it is just a figurative use of the literal meaning of the expression.
I can't say for sure, but it seems to me that it comes from the days of sailing vessels, as one of the quotes above infers, "...hang tight, good rope...". A rope that was under full load from wind-filled sails was not in any way slack or swinging. It was almost as rigid as a wooden beam. It was doing its job of keeping something in place. So "Hang tight" meant keep in place, remain steady and strong, and when the wind changes THEN you can relax. Or something like that, anyway. Just my opinion...