The expression “once in a Blue Moon” has an astronomical and calendric origin that has been the subject of much study in the recent past. Briefly, a Full Moon has been called “Blue Moon” if it is the second Full Moon in a calendar month: this was introduced in a Sky&Telescope article in 1946 (Ref.1). It was derived from a misunderstanding of a convention in the “Maine Farmers' Almanac” around 1937, to call the third Full Moon in a season that has four Full Moons, a Blue Moon.
The astronomical and calendrical background is, that usually there are 12 Full Moons in a year (so one in each month), and 3 in a season. These Full Moons have traditional names, like “Harvest Moon”. The most important one is the Easter Moon: the first Full Moon in spring; Easter is celebrated on the first Sunday after that.
A lunation lasts about 29.5 days on average, so 12 lunations count 12*29,5 = 354 days: 11 days short of a year. So it may happen that 13 Full Moons (or any other phase) fall within a calendar year, and 4 in a season of 3 calendar months: this occurs 7 times in 19 years (Metonic cycle). This irregular ominous extra (“embolismic” or “leap”) Moon apparently was called the Blue Moon.
The first known occurrence of the term is a rhyme in an anti-clerical pamphlet from 1528:
“Yf they say the mone is belewe, we must believe that it is true" (WorldWideWords site, Ref.2; also see Ref.6).
The common understanding is that it refers to something absurd, like a blue coloured Moon. The equally absurd notion that it is made of green cheese dates from the following year. A more or less definitive account of all this was given in a Sky&Telescope article in 1999 (Ref.3), on occasion of the occurrence of 2 Blue Moons that year, in January and March.
However, later the website of the Maine Farmers' Almanac (page now defunct: Ref.4) suggested an alternative meaning: "belewe" is said to also be an old-English word for "betray": this extra Full Moon was somehow deceptive (however see the discussion in this article: Ref.7). Anyway this suggests to me an origin and meaning that I have not seen explained in detail before in the discussions.