I suppose a big portion of it is what materials you use. Obviously, try to use the thinnest materia possible without compromising strength. I've seen designs using anything from .04" of versatubing to 1/8" thick box channel and thicker. I know we tend to use the heavier stuff in our drivetrains and the lighter stuff in mechanisms; generally the 1/8" box channel for drivebase, and 1/16 for mechanisms. 1/8" plate tends to be more than enough for most of our purposes; we only used 1/4" plate for hooks during ultimate ascent, and we probably could have used 1/8" there too.
The other thing is to use belt and aluminum/plastic pulleys wherever possible. You can pocket gears and still retain the strength for a smaller weight. Pneumatics substitute for motors really well in most cases; they are a good bit lighter when using many.
That's all I can think of in terms of general ideas ATM. I suppose most of the weight really comes down to how the mechanism is designed; you may want to review your design idea overall first?
And then there's always swiss cheesing ...
