This question seems slightly strangely phrased to me: the case $\mathbb C[B_n\backslash GL_n(F_q)/B_n]$ gives you a Hecke algebra which is *not* commutative. It has the same dimension of the group algebra of the symmetric group, and by deformation techniques it's known to have the same number of irreducible representations, but none of this requires commutativity (though that does hold sometimes, and David's answer about the Gelfand trick is a very slick strategy).

If you're interested in where the theory of Hecke algebras relating representations of the symmetric group $S_n$ and the representations of $\text{GL}_n(\mathbb F_q)$ extends to, then there are a number of directions. For a finite group of Lie type $G$, the classification of all irreducible representations hinges on generalizations of the kind of Hecke algebra you mention, where instead of the double-coset algebra, you consider twisted versions of it which are endomorphism algebras of certain "cuspidal" representations of (rational) Levi subgroups of $G$. These turn out to have presentations like the Hecke algebra attached to $S_n$, though sometimes with parameters which are various powers of $q$. One standard reference is Carter's book on finite groups of Lie type, where there is discussion of these endomorphism algebras. (A crucial ingredient is usually that $G$ has a $BN$ pair, so that the $B$--double cosets are indexed by a Coxeter group.)

Similar kinds of Hecke algebras (though now infinite-dimensional "affine" Hecke algebras) arise when studying representations of p-adic groups, Roger Howe has a nice article in Lecture Notes in Math 1804 on the use of Hecke algebras in the p-adic theory -- slight generalisations of the affine Hecke algebras used to study the case of representations generated by an Iwahori fixed vector turn out to allow you to study surprisingly large parts of the representation theory of p-adic groups.