Well, you are both right, in that you both learned the phrase you're still using. In terms of the original derivation of the phrase, only one usage is accurate. Sorry, Laurie, it's 'think' 
And for those who say 'think' can't be used as a noun, of course it can; its used in exactly that way in phrases such as "I'll have a think about it" or "Have another think about it" - precisely the usage in the second part of the saying.
In the 'thing' argument, I've never understood what the first 'thing' was supposed to be which the 'other thing' would follow. Obviously in the 'think' argument, you've already got your first 'think' there in opposition to the 'other think'.