When did "Space" become the accepted terminology for the area beyond the Earth's atmosphere? What, if anything, was it called before it became known as "Space"?
The English word space originates from Latin word for expanse -
spatium (also written spacium in Medieval Latin), and later French word
espace, through the use of which English variant space was formed.
The earliest mention of space in relation to the 'outer space' is attributed to the 17th-century English poet John Milton, that used it in his epic poem Paradise Lost to much the same effect we now use it to describe 'outer space' beyond the Earth’s atmosphere too thin for aeronautical purposes (or other non-arbitrary definitions):
When I behold this goodly frame, this World,
Of Heaven and Earth consisting, and compute
Their magnitudes—this Earth, a spot, a grain,
An atom, with the Firmament compared
And all her numbered stars, that seem to roll
Spaces incomprehensible (for such
Their distance argues, and their swift return
Diurnal) merely to officiate light
Round this opacous Earth, this punctual spot,
One day and night, in all her vast survey
Useless besides—reasoning, I oft admire,
How Nature, wise and frugal could commit
Such disproportions, with superfluous hand
So many nobler bodies to create,
Greater so manifold, to this one use,
For aught appears, and on their Orbs impose
Such restless revolution day by day
Repeated, while the sedentary Earth,
That better might with far less compass move,
Served by more noble than herself, attains
Her end without least motion, and receives,
As tribute, such a sumless journey brought
Of incorporeal speed, her warmth and light;
Speed, to describe whose swiftness number fails. — viii. 15-38.
Milton's description of the Universe and its celestial bodies is described in detail in The Project Gutenberg EBook of The Astronomy of Milton's 'Paradise Lost', by Thomas Orchard.
Milton is not, in fact, the first English author to use 'space' in this sense. He is antedated by Lucy Hutchinson and John Evelyn, both of whom use 'space' to translate 'spatium' in their English versions of Lucretius's De Rerum Natura. Milton probably did help the English sense to catch on, and he too was alluding to Lucretius. By the way, Paradise Lost, book 1, line 650 (which has the singular, 'space may produce new worlds') is a better example to use than the lines in book 8.