When cutting portraits, I use both 1/8" and 1/4". I found the thicker you cut, the eaiser it is to control (up to a certain point) and you have a lot more support in delicate areas. So if I was cutting 1/8", I'll stack cut 3 at a time to give me the thickness I need for control and support. This is nice too since I get 3 copies to give away. If I cut 1/4", I usually just cut one. I personally like the look of 1/4", but if you have a lot of thin lines/line drawings or a pointalism type of pattern, it may not be as effective because you don't see the backer as well.
Barry's right about finding frames for 1/8". It'll be much easier. 1/8" works really well, especially if it's behind glass (the glass keeps the wood pressed against the backer). If you won't have glass, I'd suggest 1/4", unless you decide to glue the ply to the backer.