Wordorigins.org

A rain check was originally a voucher issued at baseball games that were rained out, allowing the spectators to return to watch another game. From the St. Louis Post-Dispatch of 26 May 1884:

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.

Gradually, the usage spread to other types of sporting events and eventually to any offer that was not taken up immediately. The term dates to at least 1884. It probably crossed the Atlantic with American soldiers during the First World War; the Oxford English Dictionary contains the following citation from National Geographic, August 1919:

Even the sport-loving Britons are said to have admired and wondered at the American dough-boy...issuing occasional rain-checks in mid-inning when the downpour of bursting shells became too distracting.

There is a commonly told tale that the first rain check was issued by the minor league New Orleans Pelicans and owner Abner Powell sometime in 1887-89 (the dates vary depending on who is doing the telling). But as can be seen from the 1884 quote from Missouri, this claim is not true.

(Sources: Oxford English Dictionary, 2nd Edition; New Dickson Baseball Dictionary)