If you want to be satisfied, stick with this till the end. It won't take long. (That's what she said.) The phrase "that's what she said" isn't an ancient biblical epitaph. It has very definite origins and they're are actually fairly recent. The folks at the Today I Found Out YouTube channel have attempted, in the video above, to trace back through the history of your dad's favorite joke.
In America, the earliest documented case of the phrase appears in a 1975 episode of Saturday Night Live. In particular, it was Chevy Chase using the joke during "Weekend Update" on the show's first season. It was kept in popular use on Saturday Night Live through repeated uses in "Wayne's World" sketches and, later, the movie.