For some reason robin doesn't bring his grate brane here very often, so I passed the question on. Verbatim:

The most obvious use of an exhaustive permutation generator is brute force search. Examples aren't difficult to come up with - see for example Jason Brazile's 1998 TPJ article Iterating Over Permutations which begins "My nephew asked for help with a homework problem" and proceeds to show how a permutation generator can be used to brute force a solution to the problem.

The ideal motivating problem would be one that is:
a) Susceptible to brute force search over permutations
b) Not obviously solvable by a more efficient method
c) As natural (uncontrived) as possible.

There is an extensive directory of NP-complete problems which satisfy (b) and (c), which would seem to be a good place to start looking.

For example, the minimum linear arrangement problem obviously satsfies condition (a).

More useful, perhaps, is the problem of "Sequencing with Release Times and Deadlines" which any programmer who has to manage her own time will be informally familiar with! That can be solved by trying all the possible ordering of tasks (i.e. all the permutations of the task list) until an ordering is found which works (or not).