After dreaming about making one for a long time, I finally went ahead and did it. It was a lot of fun to make, and it worked pretty much as advertised. The biggest problem I encountered is that I was over driving the fan motor and I probably could have chosen a more powerful one that could have handled more tubing (more weight). I will solve this problem for revision two.
Anyhow, without further ado, here is how to make your own.
Step 1: Go Get Stuff
A box fan
A 1' foot metal tube with a diameter slightly bigger than your motor shaft
A rod clamp slightly bigger than your metal tube
Step 2: Take Apart the Fan
Forcibly pull the fan blade mount from the motor shaft.
Step 3: Measure
Also get a rod clamp with a set screw that can fit around the tube.
Step 4: Clamp
Step 5: Strip the Umbrella
This may require breaking, bending and/or sawing off a clamp at the top of the umbrella that is holding the cloth in place.
Step 6: Saw Off the Handle
Step 7: Remove the Spring
Step 8: Simplify
Step 9: Free It
Step 10: Remove the Extra Hangy Bits
Step 11: Fasten the Umbrella
Simple lower the bottom part onto the shaft and don't worry about securing it for now (it will be held in place later by zip ties).
Bring the top part of the frame near the top part of the shaft and drill completely through both the top bracket and the shaft. Insert a metal pin through this hole and bend it to lock the top in place. If the top is loose, you can insert a rubber O-ring to solve this problem.
For my pin, I sawed in half one of those "dangly bits" from the last step, inserted it through the hole and bent it upwards. This worked great.
Step 12: Cut Tubes
Step 13: Zip Tie
While zip tying them, you should also determine that approximate height you want the bottom part of the umbrella frame to sit at and zip tie this in place as well.
Step 14: Plug and Play
I found that you shouldn't run it for more than a few minutes as you are typically over-working the motor and it starts to get worryingly hot.
I have not had the chance to properly record this yet, but I would imagine the ideal place to have a mic would be in close, below the spinning mechanism, pointing upwards at 45 degrees.