What do you mean by "in a rigorous sense"? I would be interested to see what others think, but what you are describing seems okay to me:
The norm of a vector, in the case of a 3-D vector, like here, is its length. To normalize a vector, you divide the vector by its norm, creating a unit vector with the same direction as the original vector. So to normalize a 3-D vector, you would divide the vector by its length. Thus it seems that
\[g=f(r,\theta,z)/\sqrt{a^{2}+c^{2}}\]
would be the normalized function, as you have suggested. Again, I do not know if this is "rigorous" enough.