See mom, it's good I don't throw anything away!
In order to do this, I needed an adapter form the tripod mount to the light stand mount. I could buy one from Amazon for $12 including shipping, or I can spend half an hour at home depot wandering the halls.
The end result should look like this:
Obviously, it would be best to use a lathe, but getting one of those would push the price a little bit past $12.
Here's what I did:
Using a large washer, small washer, 1/4" coupler nut, and small 1/4" bolt, I screwed the nut onto the tripod and the screw onto the nut. That's all!
And all for less than $2, given that I only needed to buy the coupler nut. And here it is doing it's job: