Wordorigins.org

This phrase is often said to have a nautical origin involving cannon balls, but cold enough to freeze the balls off a brass monkey is instead both literal and anatomical in origin. And although now the phrase is used almost exclusively in the canonical version given here, in early usage the parts of the monkey’s anatomy varied, as did the temperature, with the phrase being used to refer to hot weather as well.

The OED2’s entry on monkey includes the following citation from Frederick Chamier’s 1835 Unfortunate Man. While not in the form we’re familiar with today, it establishes monkeys as metaphorical instruments of weather measurement:

He was told to be silent, in a tone of voice which set me shaking like a monkey in frosty weather.