According to the Naval Historical Center of the U.S. Department of the Navy, there's no evidence that there was ever a device called a brass monkey:
The first recorded use of the term "brass monkey" appears to dates to 1857 when it was used in an apparently vulgar context by C.A. Abbey in his book Before the Mast, where on page 108 it says "It would freeze the tail off a brass monkey." (Source: Lighter, J.E. ed. Random House Historical Dictionary of American Slang. (New York: Random House, 1994): 262.)

It has often been claimed that the "brass monkey" was a holder or storage rack in which cannon balls (or shot) were stacked on a ship. Supposedly when the "monkey" with its stack of cannon ball became cold, the contraction of iron cannon balls led to the balls falling through or off of the "monkey." This explanation appears to be a legend of the sea without historical justification. In actuality, ready service shot was kept on the gun or spar decks in shot racks (also known as shot garlands in the Royal Navy) which consisted of longitudinal wooden planks with holes bored into them, into which round shot (cannon balls) were inserted for ready use by the gun crew. These shot racks or garlands are discussed in: Longridge, C. Nepean. The Anatomy of Nelson's Ships. (Annapolis MD: ]Naval Institute Press], 1981): 64. A top view of shot garlands on the upper deck of a ship-of-the-line is depicted in The Visual Dictionary of Ships and Sailing. New York: Dorling Kindersley, 1991): 17.

They do add the following:
"Brass monkey" is also the nickname for the Cunard Line's house flag which depicts a gold lion rampant on a red field. (Source: Rogers, John. Origins of Sea Terms. (Mystic CT: Mystic Seaport Museum, 1984): 23.)

So it appears that the more colorful theory of the origin of the phrase "cold enough to freeze the balls off a brass money" is a legend, similar to the myth that the word fuck is an acronym.

Source: U.S. Department of Navy, Navy Historical Center: