No, zzxxee, it wasn't me who missed a bit out, it was this funny website. I typed in a full answer, but, as you can see, I was cut off in my prime. It looked as if there wasn't any more room for little old me, so, here goes. I'll have another go at posting an answer.
I'll introduce a serious and learned note into all this unseemly ribaldry. My dictionary is the Bloomsbury Concise English Dictionary, published in 2005, and this is what my dictionary says on the subject (so, sober up everyone, and pay attention): Willy: early 20th century < Shortening of the proper name William. No, I don't know why William (whoever he was) was thus immortalised, but maybe that could start a whole new series of speculations. There, I've said it!