LibertyLover says:

I suspect it may be older than we all think. A “beaver” is part of a helmet that covers the mouth. (smirk)

Other myths from the inet:

1) It evolved from trappers masturbating with the soft pelt of a beaver.

2) It is a reference to castoreum; the bitter, putrid smelling secretion produced by the male and female beaver. Which collects around the vaginal follicles of the female beaver.

I hate to say it, it is a direct remark towards the smell of an unwashed woman’s vagina. Which after quite some time, can begin to resemble this smell. It was a phrase popularized in the 17th century, when the trade of castoreum went from the Americas to Israel, and everywhere in between.

3) Question: What is a beaver?

Answer: A furry critter that sits on your face and tries to swallow your tongue.

4) In the 17th – late 19th century, women were wont to depilate their vaginal regions for both fashionable and health reasons, however for comfort purposes (using the equipment of the day) it was found that some form of covering was required to avoid irritation. The remedial object was known as a merkin. You’ll find this word in large dictionaries described as a female or pubic wig. Merkins were fashioned from fine beaver pelts, hence the modern idiom.