I see the pattern [Adj1]っちゃあ[Adj1]（けど） relatively often, and I know that I've learned it before... I even know what it means. ...but what is it an abbreviation of? It is an abbreviation of something, right?
I've thought maybe ～といえば～ or ～というのは～, or something else along those lines, but 1) none of them seem to feel right to me, and 2) there aren't any relevant results on Google with those, so I assume (perhaps incorrectly?) that they're incorrect.
So what is っちゃあ?
「(adjective)っちゃあ(same adjective)（だ）けど」＝
「～～って[言]{い}っちゃ（あ）～～（だ）けど」＝
「～～って言ったら～～（だ）けど」
"somewhat ~~", "~~ to a degree", "You could call it ~~, I guess", etc.
「[忙]{いそが}しいっちゃあ忙しいけど」= "I am somewhat busy if not terribly so".
The nuance is that you are not busy enough to want to actively tell everyone that you are busy. If anything, you are only mildly busy. Hope this explains why the quotative「って言う」 is used in the expression. You are in effect "labeling" the degree of your own occupiedness.
You are right. This (the version using っちゃあ) has been a very common colloquial expression for quite a few years now.
I do not remember hearing or saying it as a kid, but that may be because I was living in Nagoya back then. Around Tokyo right now, you would hear it many times everyday.
っちゃあ is totally convertible to といえば, except that っちゃあ sounds very chatty and rough while といえば can be used in formal expressions. You can say 忙しいといえば忙しいけど, and it conveys the same meaning.
As for the word form, actually there are other forms like っていやあ and ってやあ. They phonetically seem to fall between といえば and っちゃあ, making the presumable process of sound change traceable like this:
Plus, there is even more shortend form:
All these six forms are possible, though #4 is rather rare.
By the way there are also forms like ってえば and ってば, which seem to be another line of sound change: Qteieba → QteHba → Qteba.
