This from a website visitor:
History: The Brass Monkey - In the heyday of sailing ships, all war ships and many freighters carried iron cannons. Those cannon fired round iron cannon balls. It was necessary to keep a good supply near the cannon. But how to prevent them from rolling about the deck? The best storage method devised was a square based pyramid with one ball on top, resting on four, resting on nine, which rested on sixteen. Thus, a supply of 30 cannon balls could be stacked in a small area right next to the cannon. There was only one problem -- how to prevent the bottom layer from sliding/rolling from under the others. The solution was a metal plate called a "Monkey" with 16 round indentations. But if this plate were made of iron, the iron balls would quickly rust to it in the salt air environment. The solution to the rusting problem was to cast the monkey out of brass. Thus the "Brass Monkey." Few landlubbers realize that brass contracts much more and much faster than iron when chilled. Consequently, when the temperature at sea dropped too far, the spacing between the indentations and the indentations themselves would shrink so much that the iron cannon balls would come right off the monkey. Thus, it was quite literally, "Cold enough to freeze the balls off a brass monkey!" And all this time, you thought that was an improper expression, didn't you?

This from another website visitor:
Don’t let anybody convince you of this. It’s rubbish. There’s no evidence that such brass plates existed. Although the boys bringing charges to the guns from the magazine were known as powder monkeys and there is evidence that a type of cannon was called a monkey in the mid seventeenth century, there’s no evidence that the word was ever applied to a plate under a pile of cannon shot. The whole story is full of logical holes: would they pile shot into a pyramid? (hugely unsafe on a rolling and pitching deck); why a brass plate? (too expensive, and unnecessary: they actually used wooden frames with holes in, called garlands, fixed to the sides of the ship); was the plate and pile together actually called a monkey? (no evidence, as I say); would cold weather cause such shrinkage as to cause balls to fall off? (highly improbable, as all the cannon balls would reduce in size equally and the differential movement between the brass plate and the iron balls would be only a fraction of a millimetre). What the written evidence shows is that the term brass monkey was quite widely distributed in the US from about the middle of the nineteenth century and was applied in all sorts of situations, not just weather. For example: from The Story of Waitstill Baxter, by Kate Douglas Wiggin (1913): “The little feller, now, is smart’s a whip, an’ could talk the tail off a brass monkey”; and from The Ivory Trail, by Talbot Mundy (1919): “He has the gall of a brass monkey”. Even when weather was involved, it was often heat rather than cold that was meant, as in the oldest example known, from Herman Melville’s Omoo (1850): “It was so excessively hot in this still, brooding valley, shut out from the Trades, and only open toward the leeward side of the island, that labor in the sun was out of the question. To use a hyperbolical phrase of Shorty’s, ‘It was ’ot enough to melt the nose h’off a brass monkey.’ ” It seems much more likely that the image here is of a real brass monkey, or more probably still a set of them. Do you remember those sculptured groups of three wise monkeys, “Hear no evil, See no evil, Speak no evil”? Though the term three wise monkeys isn’t recorded earlier than the start of the twentieth century, the images themselves were known much earlier. It’s more than likely the term came from them, as an image of something solid and inert that could only be affected by extremes.