The particular "Noether's theorem" that I'm referring to is the one that appears in the calculus of variations: if the Lagrangian in a variational problem is invariant under a one parameter group of diffeomorphisms then the corresponding "Noether charge" is conserved. In textbooks on physics and the calculus of variations there are typically three standard applications of this theorem: conservation of energy, conservation of linear momentum, and conservation of angular momentum.
I'm looking for some other elementary examples. The other only examples that I've seen require understanding of quantum mechanics; I'm looking for simpler examples. Note that it is not important to me that the Lagrangian come from physics - any variational problem will do.