When I use it, I'm deliberately emulating baby talk. It doesn't really mean anything different than "okay" (pronounced properly); it's just a way to be, I dunno, cutesy.
I didn't know about the Little Rascals/Buckwheat connection, possibly because I grew up without a TV. I'm sure Buckwheat contributed to at least some of the popularity of "otay", and possibly I absorbed it from someone who (unbeknownst to me) was imitating the TV show, but it's also possible that this is something that each generation/school/user invents anew.
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'').
Best Answer
A number of sources agree on the date in which it was apparently first used, that is 1862.
(Etymonline)
From the Routledge Dictionary of Historical Slang and the Dictionary of Slang and unconventional English by Eric Patridge:
Reach-me-down :
It appears that hand-me-down has a later origin:
(The Dictionary of American Slang, Fourth Edition)