My 10-yr-old son says, saying "on accident" instead of "by accident" makes sense because you would also say "on purpose". I first heard "on accident" by friends from the Eastern United States. They also would say, "the floor needs swept" instead of "the floor needs to be swept". Made me crazy.
It does sound a lot more palatable once you point out how it parallels 'on purpose.'
I don't feel as bad about 'the floor needs swept' if verbal- It's obvious enough that it's short for the other form (and there's nothing illegal about shortening sentences.)
Now this is a weird thought: The whole history of any language must be a series of head-shakings and hand-wringings to new words and usages. How else can any language develop ?
My wife used to say "on accident", which I, too, took to be a natural mistake extrapolated from "on purpose". "By accident, on purpose", I used to exhort. She's now my ex-wife, I'm sorry to say—for other reasons, I'm assured.
When my family moved from Minneapolis to Pittsburgh, back in the late '60s, we discovered the second construction kc mentions. The example I remember best is from high school, an obstreperous "Hey—you want punched?". But it could take almost any form: the floor needs swept, the dog wants fed, the car needs fixed etc.
Some local oddities bother me; that one never did. I didn't retain it when we moved on, as I will with some expressions (for example "y'all"), but I remember it with some fondness.