On closer examination of the QuickCam Messenger, I could see how to disassemble the base. Once disassembled I discovered that there was enough room for a 1/4"-20 nut.
Here is how I modified the QuickCam Messenger so I can mount it on a tripod.
Step 1: Tools and Materials
1. Logitech QuickCam Messenger or another webcam with a suitable base
2.1/16", 1/8", and 1/4" drill bits
3.1/4"-20 hex nut
5.Hobby knives with chisel blades
6.Epoxy Glue or Cyanoacrylate Glue (a.k.a. "Krazy Glue") or Hot Glue
7.Isopropyl (Rubbing) Alcohol for cleaning up epoxy glue