A few basic concepts to start with. One, all aluminum is about the same stiffness (force required to flex it a little but not so much it doesn't return to the original shape when you let go), the differences are in weldability, yield strength, and bend radius. 6061 is the most common, welds nicely, has a good yield strength (how much force is needed to permanently bend it), but the minimum bend radius is at least 5 or 6 times the sheet thickness to avoid cracking. The common alloy to bend is 5052, which has the same stiffness, about half (depends on exact spec) the yield strength, also welds nicely, and has a minimum bend radius of about 1.5-2 times the sheet thickness so you can make nice tight bends.
Next, given the material the final stiffness is determined by the shape, and here height perpendicular to the direction of bending is everything. The stiffness goes as the height * height * height, or height cubed, so twice as tall is 8 times as stiff. If you have a keyboard sized sheet of aluminum you can either keep making the material thicker and thicker until it gets stiff enough for you (it generally works out that stiffness requirements result in designs that have way more than enough yield strength, which is why I'm focusing on stiffness), or you can take a sheet of thinner material and turn the edges up to make a three sided box (rear and sides turned up, front open so there's no edge sticking up to hit your hands on). Using thinner material and turning up the edges results in a much stiffer and lighter tray. You can turn the edges up or down but since you wanted to minimize the overall height I'd use some 5052 that is roughly 1/16" thick and turn the back and sides up so that they are at least 1" tall. Remember that height cubed? Turning the edge up 1" versus only 1/4" is a difference in stiffness of (1/.25)^3=64, and the tray still is only as tall as the keyboard. Only thing left is to mount it, and that's left as an exercise for the student :-) :-) (actually I can't remember your description of how you wanted it mounted, sorry).
Or if anyone knows a better forum for this, please let me know.