Suppose $P$ is a simple convex polyhedron with $n$ faces. Euler's formula and the handshaking lemma tell us that the number of edges $E=3n-6$. By a deformation of $P$, I mean a polyhedron, combinatorially equivalent to $P$, resulting from slightly perturbing the $n$ planes defining $P$. (Since $P$ is simple, any sufficiently small perturbation of faces preserves the combinatorial type.) Let's say two deformations are the same if there is a Euclidean isometry taking one to the other. A perturbation of each plane is given by 3 parameters (normal and position), and the Euclidean group is 6 dimensional. I believe this means that the space of deformations of $P$ is $(3n-6)$-dimensional—that is, $E$-dimensional. So, a deformation of $P$ is determined by one real parameter for each edge of $P$.
What could this parameter possibly be? Clearly not dihedral angle, as scaling is a non-trivial deformation which preserves all angles. Taking $P$ to be the prism of a rhombus, as suggested in this answer, demonstrates that it can't be edge length. But these two quantities are the only geometric properties of an edge that I can think of. Perhaps this is just a complete coincidence? Or perhaps my dimension argument is wrong.
Simple polyhedra might not be uniquely determined by the lengths of their edges, but simplicial polyhedra (all faces are triangles) are. This follows essentially from Cauchy's rigidity theorem.
The polar dual $P^\circ$ of your simple polyhedron $P$ is simplicial, and each edge of $P$ corresponds to an edge of $P^\circ$. If you assign to each edge $e\subset P$ the length of the dual edge, then this allows for a unique reconstruction. Simply reconstruct the dual from the edge lengths, and then dualize to obtain $P$.
There is one subtlety, in that the dualization process depends on the position of the polyhedron relative to the origin. So let us choose a conanical point for dualization, say, the average of all vertex coordinates. Put this point into the origin and you get a unique correspondence between $P$ and $P^\circ$.
The real number that you assign to each edge of $P$ in this way is not "obvious" from a visual inspection of the polyhedron, in contrast to edge length or dihedral angles. It not only depends on the edge $e$, its location or the incident faces, but on a full tubular neighborhood of $e$. But as you have seen, this complexity would have vanished if you instead would have considered simplicial polyhedra.