Definitions from Wiktionary, Creative Commons Attribution/Share-Alike License
n. A stuffed animal, blanket, or toy that a small child is more attached to than any other, and often sleeps with.
n. A high hop that a rabbit may perform when happy.
v. To perform a high hop, as when happy.
Sorry, no etymologies found.
It is in regular use by millions of parents and children, is a nickname for a few cheerful people, and it's is just not defined in any major dictionary.
Binky is a brand name of pacifiers, so that explains that source for it being in common use, similar to xerox, kleenex, etc. And, then again that is just marketing cooping a common word used in child-rearing; my binky was a blanket.
Is it from some children's book that is in the English language? Obviously the book involves rabbits! Why isn't it more common in dictionaries? The word has 5.96M hits in Google, so the word is common.
The word "bink" means a bench, or in slang a sharp dressed man. So as a root that's not helpful, except perambulators (containing baby(s) and presumably at least one binky) and sharp dressed men can be found in proximity to park benches, but that is a very loose connection. And then "y", well, that is my question?