Well, this is a phrase where the original was intentionally incorrect, grammatically. Whereas "thing" fits grammatically and makes sense. I'd never heard or read the phrase with "think" instead, until recently.
It's not unheard of to use a word as a noun. It's like saying, "Don't I get a say in the matter?" I think that's probably the closest example to "another think coming." There are also things like, "Giving the dog a wash," and, "Going for a drive," though I think that those words are designed to be nouns as well.
Although, according to dictionary.com, "think" is also a noun, so it's not wrong at all.
And I do feel that it makes more sense to say, "you've got another think coming," since you start off with, "if you think that way…" So why wouldn't you follow it up with "think"? "Thing" is just so vague, while "think" is really focusing on what was said earlier.
I'm trying to disabuse people of spelling, "Just desserts," but that is a quixotic attempt. But now I'm in the mood for some French silk pie.