"Recently popularized by the HBO "Sopranos" series (the mob-owned strip club in the show is called The Badda Bing), "badda boom, badda bing" seems to have entered the popular vocabulary in the early 1970s, possibly via one of the "Godfather" movies. Often heard in the order "badda bing, badda boom" (who's to say what's correct in such matters?), the phrase is a way of saying, "That's it, it's all taken care of" (as in "We set the Judge up with a hooker, Louie busts in with a camera, and badda bing, badda boom, case dismissed"). The exact origin of "badda boom, badda bing" is a mystery, but the best guess I've heard is that it started as a verbal imitation of a "rim shot," the drummer's punctuation to a comic's punchline in the old vaudeville routines."
It's for sure an onomatopoeic word and i completely agree with that origin stated above :)
But i'd ask some old persons here in sicily... we had similar onomatopoeic and more ancient than 70s... and the Sopranos are about a sicilian family... or not? ;)