What it says on the tin. I've always used the phrase 'in the limit of all things' but hearing '$\omega$-limit' in a chaos theory class has me wanting to use the term. That said, I'd feel really awkward using it pseudo-artistically without knowing its history.
Could someone please enlighten me regarding how this term came into being (i.e. what field of mathematics, approximately how long ago, etc.)?
I think ultimately the mathematical usage can be traced back to the Apocalypse of St. John, also known as the Book of Revelations in the New Testament, and,other classical/ancient sources. (If I am not mistaken, the phrase "Alpha and Omega" was used, outside of the Apocalypse, in the ancient world to denote the beginnings and endings of all things; the religious writers adopted it from a broader context.) Of course those usages were taken in turn from the observation that $\alpha$ and $\omega$ are the first and last letters of the Greek alphabet. The usage of $\omega$ to denote the ultimate end point of any given situation has thus been around since antiquity, I believe; it crept into the common tongue and was imported into mathematical terminology from there. My take on it, in any event.
Hope this helps. Cheerio,
and as always, Fiat Lux!!!