The Oxford Dictionary of English Etymology (ODEE) says about the verb wrap (with the abbreviations expanded): “…of unknown origin, similar in form and sense are North Frisian wrappe stop up, Danish dialectal vrappe stuff; and cf. Middle Engl. bewrappe, beside wlappe (XIV), LAP3.” XIV means “the 14th century,” and LAP3 is a synonym of wrap (as in overlap), related to lap “front part of a skirt,” which has a solid etymology. The quotation from The ODEE repeats what can be found in the OED. For “cf. wlappe” some dictionaries offer the dogmatic, unsupported statement that wrap is a doublet (so Skeat) or “corruption” of lap.
So once again we encounter the off-putting formula “origin unknown.” But, as we have seen more than once, in etymology, “unknown” is a loose concept. Minsheu, the author of the first etymological dictionary of English (1617), cited two words, which, in his opinion, could be akin to wrap. One of them was German raffen “to pile, to heap.” In 1854 the same idea occurred to a certain D.B., a contributor to Southern Literary Messenger (Richmond, Va.), the author of a four-page article on word origins. Like Minsheu, he also cited two probable cognates. To both authors’ second candidates we will briefly return below. D.B., I assume, was not a famous researcher, and Southern Literary Messenger is not everybody’s regular source of information on linguistic issues (again a mere guess). But in 1904 Heinrich Schröder, contrary to the semi-anonymous D.B., a distinguished, even brilliant German scholar, published a very long article in a leading philological journal and, and among many other things, proposed Minsheu and D.B.’s etymology as his own. Like Cato, who never stopped rubbing in his appeal Carthago delenda est (“Carthage should be destroyed”), I will keep repeating that we need summaries of everything ever said about word origins in any given language and only then shed words of wisdom to the public eager for reliable information. In the absence of summaries and surveys, etymologists, naturally, hit on the same, seemingly attractive hypotheses again and again, without realizing that the wheel has already been invented and even reinvented more than once.
Thus, three people suggested the affinity of wrap to German raffen. But not a single German dictionary I have consulted contends that raffen is akin to wrap, though, obviously, if A is related to B, B must also be related to A. This is another curious comment on the state of the art. While working on the entry raffen, the authors of German etymological dictionaries never thought of looking up wrap. And why should they have done so? In their sources, the comparison does not occur, and no one alerted them to the fact that in the English-speaking world some etymologists had tackled their word.
Closer to home than raffen is Engl. warp, whose original meaning was “to throw,” as evidenced by Dutch werpen and German werfen. Time and again, beginning with Minsheu, it has been said that wrap is a metathesized variant of warp. However, when a word falls victim to metathesis, which is a mechanical phonetic change not caused by semantic factors, the new form, with the sounds transposed (and this is what metathesis is all about), it continues meaning the same as its “parent,” while “to throw” and “to enfold” do not look like even remote synonyms. Another putative etymon of wrap that appears in some old sources is rap (so, for instance, in D.B.’s article). It is unclear which of several verbs spelled rap is meant. Perhaps it is rap “to give a quick blow, etc.,” still dimly recognized in the archaic phrase rap and rend? But that rap seems to have once had h before r, while in wrap, w- is genuine. The initial group wr- was simplified in southern English and subsequently in the Standard only in the seventeenth century, so that the fourteenth-century spelling of wrap inspires confidence. The other rap “to strike” may be a sound-imitative verb of Scandinavian origin, and neither h- nor w- has been recorded in it in any Scandinavian language.
Other conjectures are even less appealing. For instance, Old Engl. wrion “bend, contort” (its reflexes are hidden in Modern Engl. wry and wriggle) is phonetically too remote from wrap. Among several look-alikes, attention has been called to the short-lived and rare late Middle English words wrabble “to wriggle,” wrabbed “perverse,” and wraw “to mew.” The last of them is obviously onomatopoeic, like mew, moo, and the rest. The original sense of the Modern English verb warble was “to whirl”; hence “to sing with trills and quavers.” This verb had hw- at one time. The enigmatic wlappe “wrap” is said in the OED to be apparently a blend of the verb lap and wrap. But so little is known about most of those words that their mutual ties cannot be reconstructed. It only seems that at least some verbs beginning with wr- and hr- were sound-imitative and possibly sound symbolic. One of them was Greek raptein (historically, with initial h-) “to stitch together,” whence rhapsody (“the stitching of songs”), in English from Greek via Latin.
The initial sense of many such verbs was “to bend, twist, stitch; wriggle; *fold, *connect, *cover.” They crossed borders with ease. Engl. wrap may be a borrowing from Frisian. If it is so, we are left with the question about its origin there. The root of the Romance verbs that have become Engl. develop and envelop seems also to be lost in obscurity, but it is characteristic that envelop sounds somewhat like lap and means approximately the same as overlap. If all of them are “fanciful” sound symbolic formations, the similarity causes little surprise. Lap in lap up is obviously sound imitative.
What then is the summary? The verb wrap appeared in English in the middle period. It has a cognate, almost a twin, in Northern Frisian. Perhaps it was coined long before it surfaced in texts, at a time when Frisian and English were closer than in the fourteenth century, but borrowing in either direction cannot be excluded. Wrap is not a doublet of warp. Nor does it have direct ties with rap in any of its senses. German raffen is hardly related to it either. Wrap has no ancient (“Indo-European”) heritage. It looks like one of a sizable number of words beginning with wr- and wl- meaning “bend, twist.” Rare Middle Engl. wrabble and wrabbed are, most probably, related to it. Without certainty, warble can be added to the group. We have no way of knowing whether Middle Engl. wlappe had an independent existence or was a blend of lap and wrap. Some words structured like rap and lap (without initial h- and w-) were close to wrap, so that confusion between and among them was possible.
Despite all the caution required by such a hard case, it can probably be stated that wrap is a sound symbolic verb. The evidence at our disposal is meager and inconclusive, but I will repeat what I have said so many times: too often the verdict “origin unknown” fails to do justice to the words we discuss. Sometimes there is indeed nothing else one can say (notably so while dealing with slang), but a verdict that presupposes death sentence should not be returned without serious deliberation.