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.):
But if we did, then we have another thing coming, for this is the cry-baby talk I find in this morning's (Dec. 16) editorial
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:
'Another thing coming' is just a mispronunciation of the original phrase. The source of this mistake is probably the duplicated 'k' sounds of 'think' and 'coming'.
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)...
The original expression is the last part of a deliberately ungrammatical joke: “If that’s what you think, you’ve got another think coming.”
Here's an instance from The Leather Workers' Journal, 1904, showing it's been around a long time...
If the wage earner thinks that he will obtain anything from either of the old parties he has another think coming.
EDIT: As noted in @Barrie's comment below, the noun usage of a think to mean an act of thinking is long-established ("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.
"Have/got another thing coming" dates back to at least 1897:
They imagine that these battles and quarrels of the track are carried on after the races are over. The people who think this ‘have another thing coming’, for the men travel in one of the most peaceful parties that follows any line of sport.
Elmira Daily Gazette & Free Press (New York)
According to the OED, it's "arising from misapprehension of to have another think coming". From Merriam Webster we know that this version dates back to at least 1867:
"By Jove, Ned!" he exclaimed. "I believe I have it." "Have what—a fit of seasickness?" "No, but these empty seats—the persons we saw you know—they belong there and they're afraid to come out and be seen." "Why should they be—if they're not the Fogers. I guess you've got another think coming." "Well, I'm sure there's something mysterious about those two—the way they hid their faces as they came on board—not appearing at supper—I'm going to keep my eyes open."
— Jane G. Austin, Outpost, 1867
I have never heard a learned person say another thing coming. The expression is a word play. If you think it's not you have another think coming. It makes perfect sense. If you say "If you think that you have another thing coming" what would it mean? What and where is this mysterious thing of which they speak? Many people think it's supposed to be "thing", but many more KNOW that think is the correct usage. Using thing is like some wannabe smart people that say "as best as I can" instead of "As best I can". They believe it makes them sound knowledgeable, but actually has the opposite effect.