This is just speculation, but another meaning of *check* is "to stop":

> **check**, *n.*  
> 1. An action or influence that stops motion or expression; a restraint: *Heavy rains were a check on the army's advance.*  
> 2. The condition of being stopped or held back; restraint: *kept my temper in check; holding agricultural pests in check with sprays.*  
> 3. An abrupt stop in forward movement or progress; a halt.

Some examples:

 * "In order to understand how the falling [rain checks][1] the wave motion,"

 *  "abundance of [rain checks][2] the propagation of cholera"

So, it could also be said that rain checks play. Perhaps the issuing of the paper checks (cheques, tickets) was also a pun on this meaning.

Anyway, I couldn't find any 1880s citations or earlier, but found some 1890s: *Outing: Volume 24* from [1894][4] and *Chimes from a jester's bells; stories and sketches* from [1897][3].


  [1]: http://books.google.com/books?id=smDnAAAAMAAJ&q=%22rain%20checks%20the%22&dq=%22rain%20checks%20the%22&hl=en&sa=X&ei=yJAzT56sNun14QT21KjmAQ&ved=0CFQQ6AEwBg
  [2]: http://books.google.com/books?id=c4JEAAAAMAAJ&pg=PA71&dq=%22rain%20checks%20the%22&hl=en&sa=X&ei=yJAzT56sNun14QT21KjmAQ&ved=0CFwQ6AEwCA
  [3]: http://www.archive.org/stream/chimesjester00burdrich#page/n65/mode/2up
  [4]: http://www.archive.org/stream/outing24newy#page/80/mode/2up/search/%22rain+checks%22