I would go with whatever is cheapest per inch of height. If the prices were all equal I would go with the 2x8s. Second choice would be 6x6 and last place would be 5x5. Don't really care for less than 6" in height for a raised bed system. Will you have open ground underneath? That would make a difference, but I would still, personally, not feel OK with less than 6" even though it would probably be just fine.

I have attempted to get two pieces of lumber stacked together in order to use them in a raised bed and it wasn't easy. I scrapped the idea. In my case it was landscape timber which I guess is about 3x4. First, 5" is a lot to drill through unless you have good quality tools and are used to doing such things. Second, there is no guarantee they will not seperate under load over time.

2x8s would be the easiest to work with, if I went with the 6x6 I would probably reinforce with brackets on the corners and every cross beam. With 2x8s I just run a screw through one board into the other.

the 6x6's are $10.50 and the 2x8's are $12.80. both cedar.

I'm digging the sod off the area where the beds will be.

Well, you have to determine your own budget, but I would personally go for the 2x8s. Also, save your back and don't strip the sod.

6" of soil on top of sod will kill it. In case you have some mutant grass that can grow through 6" of soil on top of it, just lay 6-8 sheets of newspaper over it before putting the frame down. Overlap generously. The newspaper acts like a weed mat in that it will smother the grass, but it is organic and will be worm/bacteria/fungi eaten not long after the grass is dead and will not interfere with root growth for the plants.

I mean, strip the sod if you really want to, but it is unnecessary labor. When I make new garden areas these days I don't do formal raised beds with frames anymore. I just edge the area (in ground edging to prevent grass roots from creeping in), scalp mow the lawn, lay the newspaper, cover with an inch or two of compost, plant through the news paper and then mulch with shredded wood. No grass grows through. If the area is a wetish area then I bring in topsoil to mound several inches. I am not telling you to not make a framed raised bed, there are many reasons to do this. Rather I am simply saying that the newspaper method doesn't require much of anything on top of it to kill the grass.

I think the 2x8's would be easier to work with. I agree with username on using the newspaper to kill the weeds- cardboard works too!