Google's ngram viewer allows a comparison of the frequency among two or words or phrases between selected years.
The following graph plots the frequency of "flat tax" vs. "progressive tax" during 1960-2008. (Click on link) It shows that "flat tax" took off around 1980 with the publication of the Hall-Rabushka plan that was the basis for many of the flat-tax bills proposed in Congress during 1982-86. It faded somewhat after passage of the Tax Reform Act of 1986, which resulted in two rates of 15% and 28%.
The frequency of the words "flat tax" accelerated again in the early 1990s with proposals from House Majority Leader Dick Armey and Republican presidential candidate Steve Forbes. The early 1990s also witnessed first wave of flat-tax legislation in the Baltic states of Estonia, Latvia, and Lithuania.
Its frequency in books declined again after 1998, but remains comfortably ahead of "progressive tax" through 2008, the latest year for which Google books data has been compiled.