Schrödinger's equation and spin quantum number
Good question.

This was one indication that there had to be something more than just the Schrodinger equation. Another problem which was perceived quite quickly is that the Schrodinger equation treated time differently than the spatial dimensions which is not in the spirit of special relativity. Thus, Paul Dirac constructed what we now call the Dirac equation. When you solve the Dirac equation in a Coulomb potential, spin, and hence the fine structure of the hydrogen atom energy levels in an electric field, falls out naturally.

This proves to be fairly technical and I won't try to do any of the mathematics for this (I don't completely understand it all), but you can find solutions to the Dirac equation for some common cases here.

As orthocresol points out, we first observed the intrinsic angular momentum of the electron in the Stern-Gerlach experiment, so people guessed what this would look like at first (this is how it was used in the Schrodinger equation) and Dirac came along and explained that people had guessed correctly. This can also be seen from the fact that the Dirac equation reduces to the Schrodinger equation in the non-relativistic, low energy case.