From http://answers.yahoo.com/question/index?qid=20100225170450AAnXSLO:

[In] baseball ... when a game was rained out, those who had tickets for that game were given a rain check which they could redeem at another game. ... The use of the term as early as 1884 gives some indication as to the popularity of baseball in the U.S. even at that time: 'The heavy rain yesterday threw a damper over local operations. At each of the parks the audience had to be content with three innings and rain checks.' (St. Louis Missouri Post-Dispatch 26 May [1884?])