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.
However, the joke is a twist of a much older British phrase that tracks back more than a century to some point in the Edwardian period (1901-1910). There, the line was "as the actress says to the bishop," in reference to actresses — whose company could be purchased after performances — confessing their sins to clergymen. It was used the same way as "that's what she said," highlighting an unintended double entendre.
That phrase has been used with some frequency in film and literature, including in a test reel for Alfred Hitchcock's Blackmail, where it was turned into "as the girl said to the soldier." That version is considered to be the first recording of a "that's what she said" joke.