Learn English – the origin of the phrase “to take a rain check”

According to this site, this term comes from rugby, where, after an infraction of rules, forwards from each team face off and push against one another.

Your source World Wide Words makes the point that the phrase right as... has appeared in many forms over the years and that right as rain probably became the favoured variant because of its pleasing alliteration. It seems like a reasonable conclusion and all I can do here is add further support to his theory from an earlier example.

This is from In the Midst of Alarms by Robert Barr, 1894:

"To whom are you engaged? As I understand your talk, it is to Miss Bartlett. Am I right?"

"Right as rain, Renny."

Perhaps it was a phrase that was already in common usage at this point but the triple alliteration here is striking.