The Oxford English Dictionary says that moon was not originally referring to month in honeymoon. They write that the word means:
The first month after marriage, when there is nothing but tenderness and pleasure’ (Johnson); originally having no reference to the period of a month, but comparing the mutual affection of newly-married persons to the changing moon which is no sooner full than it begins to wane; now, usually, the holiday spent together by a newly-married couple, before settling down at home.
So it is moon because it refers to the phases of the moon, not because it was a period of month. The first attested written use is in 1546:
1546 J. Heywood Prov. (1867) 14 It was yet but hony moone.
The World Wide Words agrees with the OED, and says that the stories about couples eating honey are actually false:
There are many invented stories about the origin of this word, mostly so sickly that I cringe at repeating them. There is, for example, the suggestion that at some time in some place there was a custom for newlyweds to drink a potion containing honey every day for the first month after the nuptials. But the word only turns up in English in the middle of the sixteenth century.
Let me quote you a passage from Richard Huloet’s Abecedarium Anglico Latinum of 1552 (in modernised spelling): “Honeymoon, a term proverbially applied to such as be new married, which will not fall out at the first, but the one loveth the other at the beginning exceedingly, the likelihood of their exceeding love appearing to assuage, the which time the vulgar people call the honey moon”. Putting it simply, it was that charmed period when married love was at first as sweet as honey, but which waned like the moon and in roughly the same period of time.