The most common answer is that the full moon appears blue, and to some people, when a full moon appears twice in the same month, the second one is "clearly" bluer than the first one. Another story is that it originated as a mockery of a person who insists he always tell the truth. You mock him by saying "Indeed, if you say the moon is blue, or it is made of green cheese, why, we must believe it".

My dictionary of idioms says this: "The allusion to the moon being blue goes back at least to a 1528 rhyme: Yf they say the mone is blewe, We must beleve that it is true." Etymonline gives: "Blue moon emblematic of 'very rarely' suggests something that, in fact, never happens (cf. at the Greek calends), as in this couplet from 1528: [same as above]" The Guardian's Notes and Queries colum has some nice responses as well: http://www.guardian.co.uk/notesandqueries/query/0,,-5969,00.html

The most obvious meaning of blue moon is when the moon (not necessarily a full moon) appears to a casual observer to be unusually bluish, which is a rare event. The effect can be caused by smoke or dust particles in the atmosphere, as has happened after forest fires in Sweden in 1950 and Canada in 1953 and, notably, after the eruption of Krakatoa in 1883, which caused the moon to appear blue for nearly two years. http://en.wikipedia.org/wiki/Blue_Moon The moon does occasionally turn blue as a result of smoke from forest fires or particles from a volcanic eruption. Since these blue-looking moons were rare but did happen from time to time, the phrase "once in a blue moon" was coined, meaning that an event is unusual, but can happen occasionally (Kibbey). "Blue moon" has also been used as a symbol of sadness and loneliness. That appears to have a history of its own among musicians and songwriters (Kibbey). http://scienceworld.wolfram.com/astronomy/BlueMoon.html Those rare occasions when the moon takes on a markedly blue cast. Usually, when the moon (or sun) is seen low in the sky or even high in a polluted one, it is yellowish or reddish. This is a consequence of the greater extinction of short wavelength radiation by small particles and molecules. Yet, there are particle sizes for which extinction is greater for long wavelengths, and the moon seen through a sky populated by such particles is bluish. It is rare that a large population of the particles of the appropriate size (and only that size) are produced, so the blue moon is very rare: 1883 (Krakatoa), 1927 (a late monsoon in India), 1951 (forest fires in Alberta, Canada). Because of the rarity of blue moons, the phrase, “once in a blue moon,” has been used metaphorically since the midnineteenth century to denote a real, but rare event. 2. In recent times, often used incorrectly to mean the occurrence of a second full moon within a calendar month. This strange new use of the term, popularized by a board game in the mid-1980s, arose from a mistake made by the author of a magazine article written in the 1940s. Since two full moons in one month occur fairly regularly, it should not be considered a rare event. http://amsglossary.allenpress.com/glossary/search?p=1&query=blue+moon