Where does "can't be arsed" come from?

To make an effort to do something, you have to "get off your arse"!

Hence "can't be arsed" means don't feel it worthwhile to make the effort of standing up and doing something.

It is also heard in Britain.

It may be more the structure than the word "arse" that is at the origin of this. There are a few expressions where you can use a whole range of words, but the important thing is the structure. For example, you can put pretty much any past participle in the following and it will be understood to mean either "he was extremely drunk" or "he was extremely drugged" (or one or two other notions such as "tired"-- the point is there are a narrow range of interpretations compared to 'all the past participles in the universe'):

He was completely ...ed.

Or:

He was one ... short of a full ... .

with any pair of words that semantically go together will effectively be taken to mean "he wasn't very intelligent".

So similarly with "arsed", you can insert a wide variety of words (many of them expletives): "bothered", "fucked", "frigged", "buggered", "shagged". Indeed, if you say "I couldn't be jingled to do it" it would probably be interpreted in the same way. So it's not clear that there's some special meaning attached to "arsed" in this case.

We've always used arse rather than the American ass, but this expression is a recent British import. I'd actually be mildly surprised to hear it from a New Zealand-born person, and would suspect that they were consciously copying a phrase from a British comedy.