If you take off the glove box you'll see the perpetrator - a shiny, noisy, metal box.

I'm sure there are many ways you could possibly quiet this annoyance but my method seems to work.  I had some thin wire laying around that I bought from Home Depot like this:

I wrapped the wire up over a metal tube in the dash and down under the metal box like so:

While pushing up on the box, I twisted the ends of the wire together to hold the box still like so: