Deane: the variational derivative is defined on the space of functions on jet space more or less as a jet space-analog of the functional derivative, but without the actual functional aspect (as described above and below). In short, it is important in for example the geometry of jet spaces (in particular in the horizontal cohomology, where it comes from the de Rham differential along the fibers); and in mathematical physics, since one can express Euler-Lagrange equations in terms of it.

Thank you, Igor, for this answer. It was not precisely what I was looking for, but that's to be expected because I don't think I managed to write down exactly what I was looking for.