I am an older person too and I've only heard the 'thing' version until very recently. I thought when I first heard it that the 'think' version was a pun on the original 'thing'.

I'm 56 and up until last week I'd never heard the "another think" version. I thought it was a recent and clever play on words. Obviously it is not recent at all. For those struggling with the logic of "another thing" here's what I have always understood.

The expression really needs to be looked at within the context it is being used. In actual context the words "that" and "thing" refer to specific future events nearly always involving a threatening situation. Here's an example of what I mean.

"Your time's up, Jones," Smith cried, waving his gun. "On the count of three I'm gonna blow your head off. I'll be free of you bossing me around at last."

"If you think that you have another thing coming, you rotten little...."

Here "that" refers to the event or the "thing" (freedom from Jones) that Smith is thinking of / anticipating. "Freedom from Jones" in this context is the first "thing" and gives logical sense to use of "another thing". When Jones retaliates with "you have another thing coming" the meaning is that Jones will stop Smith from killing him and gaining his freedom, and that events or things are not going to work out the way Smith thought.

In other words I've always understood it to mean "If you think things are going to work out that way, you're wrong. Things are going to work out differently."

This is probably the same understanding that Bill S mentioned above.