The more I hear sentences where "you've got another think/g coming!" the more I am convinced that it is, "then you should have second thoughts." but "another think" is used for the effect.
Despite the fact that "another thing coming" can appear outside the phrase, in Ngram, BE shows it as far less popular.
The same is true of AE.
In normal comparisons, were two phrases so compared. such figures would point clearly to one being the accepted version and the other the bastard son.
I suggest a look at the comparison of "I could not care less" and "I could care less" for a similar disparity.