1) If you want a Noether theorem for information, there is no such thing.
Trying to obtain it from a symmetry law, by Noether's theorem can't work, simply because information is not a quantity that can be obtained for instance by the derivative of the Lagrangian with respect to some variable. Information is not scalar, vector, tensor, spinor etc.
2) Another way to obtain conservation laws can be found in quantum mechanics. The observables that commute with the Hamiltonian are conserved. Again, you don't have an observable, in the sense of quantum mechanic, for information.
Trying to obtain conservation of information from commutation with Hamiltonian can't work, because there is no observable (hermitian operator on the Hilbert space) associated to information. Information is not the eigenvalue of such an operator.
3) The only way, which also is the simplest and the most direct, is the following: to have information conservation, when you reverse the evolution laws, you have to obtain evolution laws that are deterministic. This ensures conservation of information, in fact, they are equivalent. In particular, most classical laws are deterministic and reversible. Also, in quantum mechanics, unitary evolution is reversible, giving you the conservation of information.
I don't say that the evolution laws have to be deterministic, or that they have to be invariant to time reversal. Just that, when you apply time reversal, the evolution equations you obtain (which are allowed to be different than the original ones) are deterministic. Simplest way to think about this is by using dynamical systems. Trajectories in phase space are not allowed to merge, because if they merge, the information about what trajectory was before merging is lost. They are allowed to branch, because you can still go back and see what any previous state was. Branching breaks determinism, but not preservation of information. Old information is preserved at branching, but, as WetSavannaAnimal mentioned, new information is added. Therefore, if we want strict conservation, we should forbid both merging and branching, and in this case determinism is required.