Some theories
Wiktionary's etymology:
Unknown, possibly from Yiddish נודזשען (nudzhen, “to badger”), or possibly from Hebrew נוגות nugot (afflict) (see Eicha/Lamentations 1:3) or possibly via an alteration of nudge, matching the alteration of wedge to wedgie.
Copying wedgie?
The OED's first-known wedgie is from 1977. Whilst this is later than the first noogie in 1968, it's not a million miles away, and playground humour won't necessarily have been documented thoroughly; I'm sure both terms were in use earlier (commenter Bib remembers noogie from the mid-1950s). But it does lend some doubt on the wedge->wedgie/nudge->noogie theory.
Yiddish?
I searched Google Books, but didn't find anything earlier than the OED's 1968:
I. Horovitz Indian wants Bronx 11 Now I'll give you twenty noogies, so we'll be even. (He raps Joey on the R. arm.)
The American author Israel Horovitz was born to a Jewish family, and the play is set in New York City. This lends some weight to a Yiddish origin.
NY / knuckles?
At least there appears to be a New York origin. Etymologist William Safire received plenty of correspondence from New York Times readers who recollected its use in the Bronx and Brooklyn (one class of '47) and added:
Noting the hard g, making the word rhyme with boogie-woogie, etymologists will make the connection of noogie with knuckle, rooted in the Dutch word knook, ''bone.'' That was related to the Middle Low German knoke, and to the Middle English knockel. By the 1940's, knuckle was also a slang word for the head,'' leading to the World War II use of knucklehead as a jocular put-down.
Further evidence that the Bronx term has roots in Holland is that the transitive verb knuckle, ''to press or rub with the knuckles,'' is also known as giving a ''Dutch rub'' (causing many a victim to ''knuckle under'').
A likely origin of the word could be latin condere that means to hide/to keep safe. I've found two references pointing to that, it all gets back to that commercial site. Not a scientific source but sounds likely so I'm risking an answer.
I'll sum up briefly one short passage of this long article. A craftsman from Utrecht decided to make those little devices out of mutton intestine because of venereal deceases spreading quickly among the diplomats during the Congress of Utrecht that lasted several months. The British diplomats brought back several of those little devices with them. They started being made on a large scale and the name condom (from latin condere) given to them.
Maybe the question could be brought to history.stackexchange ?
Best Answer
Clip has been used as a noun since 1830 to mean, in the OED’s definition, ‘a smart blow, stroke’. This definition is given under the entry for clip with the core sense of a cut, originally in the context of sheep-shearing. It is perhaps a relatively short step from applying an instrument to the covering of an animal’s skin in order to remove it to applying a stick or the hand to the body of a human in order to hurt it.