I think a bit of closed-cell foam mat, as Stuart suggested Joe at zpacks might have tried, could be a good way to go. A lightweight bit of nylon rod around the edge could help stiffen the periphery. The rod could be pulled out and used for a bivy hood or tent vent and the foam mat could be unfolded and layed down as part of a sleeping pad.
I was also imagining all kinds of other, mostly preposterous ways to make it a multi-use piece of gear (in addition to the many obvious uses). A threaded opening (like a screw cap/neck from a plastic soda bottle) could be added to the apex to accommodate a water filter to make it a scoop/funnel for filtering water. If it were somewhat bowl-shaped like Zatoichi's, rather than a cone, it could be aluminized on one side and, like a parabolic mirror, it could be used for heating something (I don't know what) in strong sunlight. If it had a drawstring around the edge, the stiffener rod could be pulled out and it could be cinched up to make a bag for carrying water (or anything else). Or it could be made out of a laminated map.