This is based on the old idea that "normal" body temperature is 98.6 degrees Fahrenheit. This is exactly 37 degrees Celsius:
C = (5/9) * (F - 32) = (5/9) * (98.6 - 32) = (5/9) * (66.6) = (5/9) * (66 + 3/5) = (5/9) * (333/5) = 333/9 = 37
I'd heard before that it's actually questionable whether 37 is really the average. Alert reader Russell Schulz points out that a 1992 article in the Journal of the American Medical Association found the actual average to be about 98.2. But I won't tell if you won't. :-)
By the way, the 1992 JAMA article was based on a study of 148 people -- a multiple of you-know-what.
Here's a related article from 1996, which discusses using the dataset from the 1992 JAMA article to teach some statistical concepts.