We consider the standard first-passage percolation model on Z^d, in which each edge is assigned an i.i.d. non-negative weight, and the passage time between any two points is the smallest total weight of a nearest-neighbor path between them.  Our primary interest is in the empirical measures of edge-weights observed along geodesics from 0 to ne_1.  For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as n tends to infinity. These families include arbitrarily small L^infty-perturbations of any given distribution, as well as almost every finitely supported distribution.  Analogous results hold for e_1-directed infinite geodesics.  Our methodology is driven by a new variational formula for the time constant, which requires no assumptions on the edge-weight distribution.  Incidentally, this variational approach allows us to recover and extend a recent result of Krishnan, Rassoul-Agha, and Seppäläinen regarding the convergence of geodesic lengths.