The kanji:
水 = water
無 = none / not any
月 = month
無 means “not any”, not “of”.
So why doesn’t this mean “the month with no water”?
According to wikipedia, this 無 is used purely as an ateji, which is a kanji used for its pronunciation rather than its meaning. 無 is pronounced “na”, which sometimes has the same meaning as the possessive particle の (“no”), thus rendering the meaning something like “water’s month”.
Which makes sense, given that June is usually the time of the Japanese rainy season!