Big Bang Nucleosynthesis, another intimidating name, is just the study of how the elements ("nuclei") were made ("synthesis") in the Big Bang model.
Between about 1948 and 1953, Ralph Alpher, along with Robert Herman, George Gamow and J.W. Follin, worked on trying to figure out the ratios of the abundances of various elements in a universe resulting from a Big Bang. For example, if one started with equal numbers of protons and neutrons, what would be the ultimate ratio of Helium and Hydrogen today?
While they do not seem to have gotten all the details right, and their estimates for the temperature of the Microwave Background jumped around a little, they do seem to be the first people to imply the existance of a bath of relic light left over from the Big Bang. They were focussing on other things, but they predicted the Microwave Background.
More background on Alpher's work can be found in the very accessible book The First Three Minutes, by Steven Weinberg.