I agree, 1/2" (12.7mm) minimum unless it was a bartop control panel....
Depends on the material and the structure. The previous posters saying 1/2" is spot on, but if you really wanted to you could do some fancy engineering with stabilizers in key places adding more structure to it. Then again we're talking more wood and metal on the inside of the CP and I'm not sure if you're doing this to save weight or size.
So yeah, 1/2" or greater please.