renew v.

word-forming element meaning "back, back from, back to the original place;" also "again, anew, once more," also conveying the notion of "undoing" or "backward," etc. (see sense evolution below), c. 1200, from Old French re- and directly from Latin re- an inseparable prefix meaning "again; back; anew, against."

Watkins (2000) describes this as a Latin combining form conceivably from Indo-European *wret-, metathetical variant of *wert- "to turn." De Vaan says the "only acceptable etymology" for it is a 2004 explanation which reconstructs a root in PIE *ure "back."

In earliest Latin the prefix became red- before vowels and h-, a form preserved in redact, redeem, redolent, redundant, redintegrate, and, in disguise, render (v.). In some English words from French and Italian re- appears as ra- and the  following consonant is often doubled (see rally (v.1)).

The many meanings in the notion of "back" give re- its broad sense-range: "a turning back; opposition; restoration to a former state;" "transition to an opposite state." From the extended senses in "again," re- becomes "repetition of an action," and in this sense it is extremely common as a formative element in English, applicable to any verb. OED writes that it is "impossible to attempt a complete record of all the forms resulting from its use," and adds that "The number of these is practically infinite ...."   

Often merely intensive, and in many of the older borrowings from French and Latin the precise sense of re- is forgotten, lost in secondary senses, or weakened beyond recognition, so that it has no apparent semantic content (receive, recommend, recover, reduce, recreate, refer, religion, remain, request, require). There seem to have been more such words in Middle English than after, e.g. recomfort (v.) "to comfort, console; encourage;" recourse (n.) "a process, way, course." Recover in Middle English also could mean "obtain, win" (happiness, a kingdom, etc.) with no notion of getting something back, also "gain the upper hand, overcome; arrive at;" also consider the legal sense of recovery as "obtain (property) by judgment or legal proceedings." 

And, due to sound changes and accent shifts, re- sometimes entirely loses its identity as a prefix (rebel, relic, remnant, restive, rest (n.2) "remainder," rally (v.1) "bring together"). In a few words it is reduced to r-, as in ransom (a doublet of redemption), rampart, etc.

It was used from Middle English in forming words from Germanic as well as Latin elements (rebuild, refill, reset, rewrite), and was used so even in Old French (regret, regard, reward, etc.).

Prefixed to a word beginning with e, re- is separated by a hyphen, as re-establish, re-estate, re-edify, etc. ; or else the second e has a dieresis over it: as, reëstablish, reëmbark, etc. The hyphen is also sometimes used to bring out emphatically the sense of repetition or iteration : as, sung and re-sung. The dieresis is not used over other vowels than e when re is prefixed : thus, reinforce, reunite, reabolish. [Century Dictionary, 1895]

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.