Middle English neue, from Old English neowe, niowe, earlier niwe "made or established for the first time, fresh, recently made or grown; novel, unheard-of, different from the old; untried, inexperienced, unused," from Proto-Germanic *neuja- (source also of Old Saxon niuwi, Old Frisian nie, Middle Dutch nieuwe, Dutch nieuw, Old High German niuwl, German neu, Danish and Swedish ny, Gothic niujis "new").
This is from PIE *newo- "new" (source also of Sanskrit navah, Persian nau, Hittite newash, Greek neos, Lithuanian naujas, Old Church Slavonic novu, Russian novyi, Latin novus, Old Irish nue, Welsh newydd "new").
From mid-14c. as "novel, modern" (Gower, 1393, has go the new foot "dance the latest style"). In the names of cities and countries named for some other place, c. 1500. Meaning "not habituated, unfamiliar, unaccustomed," 1590s. Of the moon from late Old English. The adverb, "newly, for the first time," is Old English niwe, from the adjective. As a noun, "that which is new," also in Old English. There was a verb form in Old English (niwian, neowian) and Middle English (neuen) "make, invent, create; bring forth, produce, bear fruit; begin or resume (an activity); resupply; substitute," but it seems to have fallen from use.
New Testament is from late 14c. New math in reference to a system of teaching mathematics based on investigation and discovery is from 1958. New World (adj.) to designate phenomena of the Western Hemisphere first attested 1823, in Lord Byron; the noun phrase is recorded from 1550s. New Deal in the FDR sense is attested by 1932. New school in reference to the more advanced or liberal faction of something is from 1806. New Left (1960) was a coinage of U.S. political sociologist C. Wright Mills (1916-1962). New light in reference to religions is from 1640s. New frontier, in U.S. politics, "reform and social betterment," is from 1934 (Henry Wallace) but associated with John F. Kennedy's use of it in 1960.