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 the wave motion,"

• "abundance of rain checks 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 and Chimes from a jester's bells; stories and sketches from 1897.

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 the wave motion,"

• "abundance of rain checks 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 and Chimes from a jester's bells; stories and sketches from 1897.

Edit: here it is in the Baltimore American of Dec 12, 1884, in a short column headed BASEBALL REGULATIONS about "Changes made in the Rules of the American Association". If the phrase has been used in official regulations, it's likely to have been used verbally for some time beforehand.

When clubs issue rain checks, they shall not be required to pay the $65 guarantee to visiting clubs.