early 15c., remainen, "be left after the removal or loss of a part, number, or quality; survive," from Anglo-French remayn-, Old French remain- (as in il remaint "it remains"), stressed stem of remanoir "to stay, dwell, remain; be left; hold out," from Latin remanere "to remain, to stay behind; be left behind; endure, abide, last" (source also of Old Spanish remaner, Italian rimanere), from re- "back" (see re-) + manere "to stay, remain" (from PIE root *men- (3) "to remain").

Also from early 15c. as "continue" in someone's charge or possession; continue in a certain place or condition." From early 15c. in mathematics. Related: Remained; remaining.

late 15c., "those left over or surviving," from Old French remain, back-formation (verbal noun) from remanoir, remaindre "to stay, dwell, remain, be left," or else formed in Middle English from remain (v.).

The more usual noun in English has been remainder (n.), also see remnant, except in remains "a survival, relic, remaining part of something" (c. 1500), especially "that which remains of a human body after life is gone, corpse," which sense is attested from c. 1700. As "literary work (especially if unpublished) left by an author" from 1650s.

A native word would be leavings. Old English had yþlafe "the leavings of the waves," a kenning for "shore," daroþa lāf "leavings of spears," a kenning for "survivors" (of a battle).

early 15c., "remaining, left over; left behind, remaining, continuing, staying," senses now obsolete, from Old French remanant, remenant, present-participle of remanoir "to stay; be left," and directly from Latin remanentem (nominative remanens), present participle of remanere "stay behind; be left behind" (see remain (v.), and compare remnant, which is a syncopated version of this word). In physics by 1866, probably from Latin.

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]