A solid tall cone is quite similar to what you want. I guess it can be tweaked to get a polyhedra.
So a cone sitting on its circular base is maximally stable, what position do you put the cone into that is both stable, and if it gets disturbed, even slightly, it reverts to sitting on its base?
A weeble-wobble
So a cone sitting on its circular base is maximally stable, what position do you put the cone into that is both stable, and if it gets disturbed, even slightly, it reverts to sitting on its base?