I'm not sure if this is helpful for selecting your bed frame tubing size or not but if you want to compare the different tubing sizes from an engineering perspective, I ran a few quick calculations. The first column is the weight per foot of different square tubing options. I know many aren't concerned with weight since it's a HDT but I've come from a racing background so I couldn't help myself. The second column is the "Section Modulus" which determines the strength of the member relative to a bending load. For both calculations I've used 2" x .120" wall square tubing as a baseline so you can compare both the weight and strength of other sizes to a 2" x .120" wall square tubing baseline. As an example- 2" x .25" is 71% stronger compared to 2" x .120" but weighs nearly twice as much per foot whereas 2.5" x .120" is only 27% heavier per foot compared to 2" x .120" but is still 62% stronger.
I think the big takeaway is that if you're interested in increasing strength, you're always better off increasing the height of the member (or outer diameter for tubing) than you are increasing the wall thickness.