If you're square ruling, your 8x8 post could be as small as 7.5".

Your braces can certainly be a bit more narrow in the width, that's no problem (although your structural bearing will change a little bit). Your brace thickness is important though--you'll want them exactly 4" so that you can frame the mortise 2/2 (2" from layout/2" wide). I usually saw my braces an 1/8 wide, let them dry a bit, and then plane them to final thickness.