This is an attempt to respond to the incisive comments of Steve Huntsman, Tom Goodwillie, Matt Fayers, and Ori Gurel-Gurevich; too long for a comment, and not an extension of the question.
Let me suggest a model as follows. The polyhedron $P$ is oriented randomly, and then dropped from a height $h$ onto an infinite plane. Let $k$ represent some measure of elasticity between the material from which $P$ is composed and the material of the floor plane, and let $\mu$ be a coefficient of friction between $P$ and the floor. Then the probability $p_i$ that $P$ will come to rest on face $i$ is some function of these three parameters: $p_i( h, k, \mu )$. Although there seems no question that $p_i$ does vary with these three parameters, my intuition is that, within a wide range of "reasonable" values for the parameters, $p_i$ is nearly constant. I would especially expect this to hold if $P$ is "round," say, all the vertices of $P$ lie on a sphere ($P$ is inscribed). And then if all the $p_i$ were nearly constant and equal, I would declare this a fair die.
I will admit that I cannot flesh out this intuition. And even if I could, it may be that Matt is correct that "there is no meaningful mathematics to be done here." But I retain some hope.
Let me use this non-answer to display an image of a heptahedron that follows the
Diaconis-Keller/sleepless construction, but in which I adjusted the dimensions
Benoît Kloeckner's idea of equal solid angles, in this case about 1.795 steradians
per face, which is $4 \pi / 7$: