I've never seen anyone use less than 1/8". Whatever is the minimum amount that will work, I don't know. What I do know is that the wood still needs to be straight grained, or you are wasting your time and materials.
Much of it comes down to what tools/machines you have access to. Most thickness planers won't go much under 1/8", due to safety and the strips internal integrity, before it gets chewed up.
Thinner pieces can be done safely if you have a thickness sander/drum sander.
You could feasibly laminate a backing, using 2-3 thin strips, say 1/16", and get a finished backing 1/8" - 3/16" thick.