Yes think can be a noun. But the reason the phrase sounds odd and akward is that generally you have just used "think" as a verb. As in a PP's example: "If you think preparing the nursery is tiring, you've got another thing coming when baby arrives!"
'You have another thing coming' sounds much more odd and awkward to my ears. It's jarring to hear 'another thing' when there hasn't been a first 'thing'. The sentence is audibly nonsensical.
To then use "another think coming" sounds awkward because you have not been talking about having a think in the first place. You've been talking about thinking something.
This doesn't make sense. Thinking about something is the same as having a think, surely? Whereas 'another thing' refers back to something you have not previously included in the sentence.
Also while people do say "I'll just have a think..." they do not ever say "I've got a think coming".
True, but many well known phrases uses words in a way which is peculiar to that phrase. Consider 'Curiosity killed the cat', where a word commonly understood to be a noun is the personified subject of the sentence.
Regardless of the reasons why people are wrong about this phrase (which may or may not be reasonable and explicable), they are fundamentally still wrong.