The term definitely predates D&D - the term "twenty dollar gold piece" has been in use for the $20 Double Eagle and $10 Eagle coins of the late 19th century, and also the $5 gold coin, as well.
"Gold Piece" In Print
The term is used in the Lebanon Daily News, 1 Nov 1965, of Lebanon, Pennsylvania, bottom, in an advert for old coins under the left column of text (to the right of the comics)
Four gold pieces: One (1) $20.00 gold piece, two (2) $10.00 gold pieces and one (1) $2.50 gold piece.
This alone establishes the phrase "gold piece" for gold coins in routine use prior to D&D. But let us press a little further back... say, 1913? Here's a quote from the 5 August 1913 Cedar Rapids Evening Gazette, from Cedar Rapids, Iowa, page 4, top of the third column:
The five cent piece ls the day laborer of our coinage. It la the hardest working and most successful bit at money In use In these United States. The twenty dollar gold piece Is very popular and is madly sought after In the best society; the five dollar bill has millions of friends and the hard silver dollar can be found nestling In the pocket of almost every man. But none of these like the five cent piece.
We thus have established a pattern of use for gold coins of being called "gold pieces" in the press, spanning over 5 decades; clearly not a D&D origin; not even viably a wargaming origin, for 1913 is the year of the first printing of H. G. Wells' Little Wars, the first commercially released set of wargaming rules in book form.
Searching Project Gutenberg, several ebooks have it in use...
These without clear denomination prefixed:
- Pinocchio (1883, Tr. ??? )
Author: Carlo Collodi, 1826-1890
Translator: Carol Della Chiesa, 1887-
- The Younger Set (1907)
Author: Robert W. Chambers
- A Drama on the Seashore
Author: Honore de Balzac (1799-1850)
Translator: Katharine Prescott Wormeley (1830-1908)
- Tiger Cat (1938)
Author: David H. Keller
- Pâkia (1901)
Author: Louis Becke
And several with clear denomination in dollars:
Piece
Piece is, according to several dictionaries, a common term for coins in general, of whatever denomination is specified. The quote below is excerpted from the etymology online page:
piece
early 13c., "fixed amount, measure, portion," from O.Fr. piece (11c.), from V.L. *pettia, probably from Gaulish (cf. Welsh peth "thing," Breton pez "piece"), from O.Celt. base *pett-.
[...]
Piece of Eight is the old name for the Spanish dollar (c.1600) of the value of 8 reals.
Commentary
It's pretty clear that it's a generic term for a gold coin, and for several US gold coins as well. In the US, it seems to be predominantly the popular $5 coin of the late 19th and early 20th centuries, but can be used collectively for the $2.50, $5.00, $10.00 and $20.00 gold coins; The silver coins of similar values were $0.10, $0.25, $0.50, and $1.00. Note that, still to date, "2 bits" is $0.25... a reference to the not uncommon practice of breaking Pieces of Eight (Dollares, or Reals) into 8 "bits" of an eighth-dollare each... I suspect that this is the origin of the 20:1 Silver:Gold ratio in AD&D...
Best Answer
Where does TotM come from?
"The theatre of the mind" comes from radio jargon, where the term indicates the collected ability, styles, tools, and techniques by which radio performers conjure vivid imagery in their audiences' minds through sound alone. It's used to contrast radio against visual mediums like plays, films, and books, which use additional sets of tools to conjure imagery. Orson Welles' famous War of the Worlds broadcast was part of a series of radio dramas called "The Mercury Theatre on the Air," and Steve Allen quipped, "Radio is the theater of the mind; television is the theater of the mindless."
When did TotM become an RPG term?
It's difficult to say exactly when the term first jumped mediums from radio to tabletop RPGs; likely it jumped multiple times over many years in different fragmented corners of the RPG community. The phrase certainly has been around for a good while--a company called Theatre of the Mind Enterprises sold call-of-cthulhu supplements in the early 80s, and White Wolf first published their world-of-darkness LARP system minds-eye-theatre in 1993. Neither CoC nor White Wolf made heavy use of the miniatures-on-grids paradigm--and neither did a lot of early D&D campaigns, though D&D has always included rules for miniature combat.
By early 2012 the term was ubiquitous enough that Wizards of the Coast used it in dnd-5e's beta material (then dnd-next) without elaboration (eg in the Caves of Chaos beta adventure), though they did quotate the phrase while leaving phrases like "hack-and-slash" and "cloak-and-dagger" free of scare quotes. (Often those quotes indicate a recent loan phrase from another language or discipline, not unlike italicising a recently-adopted foreign word, but I'm not sure what WotC meant by it.)
What's it used for?
Today "theatre of the mind" fills the need for a distinction between play styles which use interactive physical props as a primary focus, and play styles which do not.
With the d20 System's ubiquity, the proliferation of Friendly Local Game Stores, and because early RPGs grew out of tabletop wargames, at some point it became a common assumption that visual/physical aids like models and maps are necessary to conjure and share a scene's imagery and to make sound tactical choices. A term was necessary to call out games (systems, campaigns, groups) which instead primarily use the spoken word to conjure those scenes.
That said, the fragmented nature of the RPG community means you'll quickly find three people with five conflicting opinions about what TotM means; when in the history of RPGs it's been most common; and whether it's antithetical to a grid-and-minis play approach. In particular there are folks who consider all RPGs to be forms of TotM, and that grids-and-minis are simply an assistive aid to TotM play during tactical combat.
The term as used in radio and RPGs is unrelated to similar psychological terms; those are mostly about ways to think of brain/body interactions, not about a story's semiotic tools.