The expression seems most apt when talking about US money, as in this from NewsWeek:
'There are still titans with a conscience in the 21st century—Bill Gates, Warren Buffett, and Oprah Winfrey, for instance—but some of the rich hang on to their money until the eagle grins.'
But logically it can, can't it, fit any number of contexts other than about money? Has it ever been so used? For instance, US President might use it to add drama to some rhetoric, tap tapping the presidential seal. Or even where there is nothing money nor eagle?
When was the first use?
That line comes from Nobody Knows You When You're Down and Out, a song written in 1923. I would guess that is the source. I'm not sure.
Inside the blues lyric, it does sound like some 'on the spot' invention:
Then I began to fall so low, Lost all my good friends, I did not have nowhere to go. I get my hands on a dollar again, I'm gonna hang on to it till that eagle grins.
Until you quoted that line I had no idea. But now I suddenly remember reading once that "the eagle flies", or something like that, used to be military jargon for "it's payday". (I expect military scrip had an eagle on it.) And isn't there a British idiom about scrimping on money until [some emblem on it] squeals, or complains in some way? I can't put my finger on it just now; I probably got it from Herriott. But both sayings come to mind now that you speak of "hanging on to a dollar until the eagle grins".