This term comes from baseball, where in the 1880s it became the practice to offer paying spectators a rain check entitling them to future admission for a game that was postponed or ended early owing to bad weather. By the early 1900s the term was transferred to tickets for other kinds of entertainment, and later to a coupon entitling a customer to buy, at a later date and at the same price, a sale item temporarily out of stock.
First used when baseball games were rained out and fans were issued vouchers that allowed them to return and watch another game.
Look at that mini-wristed baller in the International League of American Superheros having a bromance! He must be a Rain Check!