  A Gelfand model for an algebra is a module given by a direct sum of
irreducible submodules, with every isomorphism class of irreducible modules
represented exactly once. We introduce the notion of a perfect model for a
finite Coxeter group, which is a certain set of discrete data (involving Rains
and Vazirani's concept of a perfect involution) that parametrizes a Gelfand
model for the associated Iwahori-Hecke algebra. We describe perfect models for
all classical Weyl groups, excluding type D in even rank. The representations
attached to these models simultaneously generalize constructions of Adin,
Postnikov, and Roichman (from type A to other classical types) and of Araujo
and Bratten (from group algebras to Iwahori-Hecke algebras). We show that each
Gelfand model derived from a perfect model has a canonical basis that gives
rise to a pair of related $W$-graphs, which we call Gelfand $W$-graphs. For
types BC and D, we prove that these $W$-graphs are dual to each other, a
phenomenon which does not occur in type A.
