Which is correct: rain check or reign check?

The correct answer is: rain check.

Note the term: "rain check", which is an informal idiom using in both American English and British English; comes from the word: "rain" (referring the weather; hence "rain check"— NOT "reign check").  It refers to the concept of plans needed to be postponed to a later date because of "rain"/weather; so a "rain check", or ticket to be issued for a future time and event.
   As an idiomatic expression, a "rain check" is often referred to an actual signed form provided as a drug store/grocery store/other merchant when a sale item is out of stock on or near the last date of sale—or when only limited quantities are available. The customer can ask a clerk—or manager—for a "rain check"; which is often a company's official form that is filled out with the sale price/quantity limited, signed and dated—& usually valid for future purchase of approved item for, say, 30 days upon approval.  {Note: Occasionally some companies have a form imprinted with an umbrella and rain drops as a humorous overtone.}.
   Additionally, as an idiomatic expression, someone might change plans to do something with someone, saying: "I'll have to take a rain check on that." [This is not necessarily an offensive thing to say; but means that one needs to reschedule a planned event with another for another time in the [near] future.