What is the origin of the phrase "you've got another thing coming"? And — perhaps more importantly — is it more correct than the alternative "you've got another think coming"?
Well, the phrase was older than I expected. NGrams reports the following matches for the phrases "got another thing coming" and "have another thing coming":
Here is quote from 1906 (I think. I am still learning how Google Books works.):
As for your other question, "you've got another think coming" wasn't an established idiom as far as I was aware, but the NGrams results shocked me:
The usage and meaning seem identical to "thing" but I find it odd that I don't recall ever seeing it in print. Looking to phrases.org.uk:
Most of the other sites I checked said similar things. Namely, "think" is the correct version and "thing" is the malformation. The origin appears to revolve around someone thinking one thing but an apparent correct will be coming shortly: They will need to rethink their previous thought (and ideally arrive at the correct position this time.)
Before reading this question, I don't think I'd ever come across "you've got another thing coming". I'm convinced by what Paul Brians says in Common Errors in English Usage: The Book (2nd Edition, November, 2008)...
Here's an instance from Punch, 1853, showing it's been around a long time...
EDIT: As noted in @Barrie's comment below, the noun usage of a think to mean an act of thinking has been around a long time ("I'll have a think about that" is perfectly normal). But note that this usage only really gained traction in the past half-century.
But in OP's context a think means what someone thinks, which is a non-standard usage. For example, we say "My thinking is we should go", not "My think is we should go". That's why as Paul Brians says in the above link, a few people in recent decades have tried to "correct" the deliberately quirky/non-standard original - but imho they've got another think coming.