Freezeing the balls off a brass monkey referes to the triangle that holds a stack of cannon balls on a ship.They were made of wood or brass. If a brass one got cold it would contract and a cannon may pop off. (the boy that got the gunpowder were powder monkeys) Hence, Freeze the balls off a brass monkey!
Sorry ,cannon ball would pop off-
Cecil doesn’t delve into the etymology of the phrase, but many other people have.
The US Navy has this to say
The OED agrees
The Phrase Finder says
Welcome to the Straight Dope Message Boards, Mark, we’re glad to have you with us.
First, on “brass monkey,” you might want to check out: http://www.word-detective.com/back-r.html#monkeys and http://www.word-detective.com/111606C.html#brassmonkey06
Second, there is already a thread on this topic. Consequently, I’m closing this one and providing a link to the other discussion: http://boards.straightdope.com/sdmb/showthread.php?t=598431 This is no problem, happens a lot, but we’d prefer to have one thread on each topic. So, please go there for further discussion (the “brass monkey” and cannonballs is discussed there as well.)