I'm thinking about this right now with my new build. I was thinking maybe 3/8", but possibly 1/2" just to make it more solid. I will be going for furniture quality with my new setup.
I've put off skinning mine for a year... but I'm thinking the same thing - 3/8 or 1/2". Need to get out to the lumber yard and see what's best in the type of wood I want.