In spite of the name, several theorems are called "Lagrange's". But usually it's one of these:

Group theory:
What Noether noded above. See the proof of Lagrange's theorem for groups.
Let f be a real function which is differentiable on [a,b]. Then for some a<c<b,
f'(c) = ---------.
If this seems to you like a trivial modification of Rolle's theorem, you're right.