I just did some research on this for my presentation at this years Magic and Meaning. The "Bra Trick" as we know it today appears to have been invented by U.F. Grant and appeared on the market in 1943. In the August 1944 issue of Genii the Genii Speak column has this to say about the effect "The GENII award for the best comedy trick of the year goes to U. F. Grant for The Climax Egg Bag. Really, Grant deserves this recognition
on two counts, the second being the Brassiere Trick"
Something called the Hollywood Brassiere Trick appeared in November of 1945 and advertised itself as an "Entirely Different Bra Trick" as it used ropes instead of the traditional handkerchiefs
The only way to become a good magician is to overcome why you became a magician -- Max Maven