Where does "noogie" come from?

Solution 1:

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'').

Solution 2:

J.E. Lighter, Random House Historical Dictionary of American Slang (1997) has separate entries for noogie and tough noogies but includes first cites to each term from 1972—and specifically from New York University:

noogie {orig. unkn.} 1. Juve. an act of rubbing or striking a person's head with the knuckles, esp. as a penalty. Also as v. [First cited occurrence:] 1972 N.Y.U. student: A noogie is a kind of a punch or a jab you give someone with your third and middle finger. You do it on the forehead or the shoulder. 2. usu. pl. a testicle. Joc. [Cited occurrence:] 1985 Northport, N.Y., woman, age 33: It's better than freezing your noogies off.

In phrase:

tough noogies! hard luck! too bad! [First cited occurrence:] 1972 N.Y.U. student: Tough noogies! is something we used to say {*ca*1959} for tough shit!

All three of Lighter's first occurrence's are from New York, although the earliest instance of "tough noogies" that I found (from 1967) is from Washington, D.C. From Sherry Schultz, "Soiree: O, Heighdy Ho," in the [American University, Washington, D.C.] Eagle (April 14, 1967) includes this item:

TOUGH NOOGIES

Congratulations to Brother Dennis Apfel of ZBT on his engagement to Donna Londesman. It seems as if ZBT will have a few sisters next September. One final word to the ZBT pledges about Supremacy Day: Tough Noogies!!

This expression seems to have had some staying power, too, as it appears again in "Bulletin Board," in the [Ithaca, New York] Ithacan (April 1, 1977):

The Registrar would like to remind students that anyone considering applying for financial aid for next year should pick up an official form from the Registrar's Office on Job Hall Two no later than March 21, 1977. Applications after this date can not be considered. The Registrar would also like it known that financial aid is not available for professors. If they do not negotiate a higher salary for themselves at contract time, tough noogies.

An early instance of noogies in the sense of "testicles" appears in Thomas McGuane, The Sporting Club (1968):

People went to and fro as though in a blackout, with a rather useless air of carrying on. A portable generator ran somewhere and lightbulbs hung in the trees, swung and heaved in the breeze and threw monstrous shadows everywhere. The children were playing in the black rectangle of shadow at the end of the tent and their fierce voices came brokenly. ". . . no, you can't! . . . Eat it raw!" Then the piping voice of a little girl, "Okay for you, Billy! Now I have to kick you in the noogies!"

The word appears in a semi-nonsense form in an unidentified poem in Howard Rosenburg, Counterpoint: An Anthology of Modern Poetry (1966):

some of our nervous noogies and other / communismists will say sublusive / things like: / What, Ho!— / i thought it was their country / and we have no business over there— / but we'll issue press releases. / if the noogie element doesn't calm down / we'll / put all the nervous noogies in jail for / their protection. we'll call the jails / strategic hamburgers / there will be onions.

Unfortunately, I have not been able to find out anything about the author of this 1966 instance of the word, nor is the word at all well defined here.

Conclusion

We have instances of "tough noogies!" in the sense of "tough luck!" from 1967, of noogies in the sense of "knuckle jab" from 1968 (in the Israel Horovitz play, The Indian Wants the Bronx, cited in Hugo's answer), and noogies in the sense of "testicles" also from 1968, as well as the strange instance of noogies in an unclear, possibly idiosyncratic sense from 1966. The first occurrences occur in a chronological clump, despite having seemingly quite different meanings.

The connection to New York seems strong in most of these early instances, but the etymological puzzle remains as unclear as ever.