Some folks order thicker or insulated fireboxes so the outside stays cooler and maybe the paint finish will stay nicer, and I guess I understand that. But I've never understood ordering it that way to save on wood. In the first place, cash in hand up front should be worth more than some perceived savings in wood use, a little here and a little there, over the coming years. But more importantly, IMHO at least, the whole purpose and function of an offset stickburner is to smoke meat with wood smoke. Given that, I just don't understand spending more money to design and build it in such a way that it will smoke less by burning less wood.
Of course, you don't want something thin, cheap, or low quality, that won't hold a temp or provide you a quality experience using it. But something 1/8 to 1/4 usually does great.
I do think something thicker or insulated might be useful to counter really cold climes, but not in Fresno. I've used mine in rain, cold, sleet, and snow and never had an issue. So my 2 cents (and that's all it's worth) would be to stick with 1/4 and use the 1/2 or insulated charges for savings or other features.