I believe Pauling's book Introduction to Quantum Mechanics with Applications in Chemistry has a derivation. I don't think the book is worth buying however so look for it in a library.
The solution is too mathematical for me. I instead like how Feynman derives the eigenfunctions for hydrogen in The Feynman Lectures on Physics. The chapters before the hydrogen atom derive the rotation matrices. After that, Feynman argues that for a given total angular momentum quantum number, if you look at the wave function along the z-axis, then since in the nonrelativistic approximation you're neglecting the electron's spin, the z-angular momentum has to be all orbital angular momentum, but you can't have a z-component of orbital angular momentum if the electron is on the z-axis, so only in the m=0 state can the electron be found in the axis direction. So for any quantum numbers of total angular momentum and angular momentum about an axis, you project the angular momentum to another axis using the rotation matrices, and find the amplitude that the new state will be found with angular momentum zero about that axis. You multiply this "spherical harmonic" by something which is a function of just the radius, dump it back into the Schrodinger equation, and then it becomes easier to work with - you don't have to define the "Legendre" or whatever polynomials.
If you're familiar with the rotation matrices, then I suggest this method as opposed to the full blown mathematical approach.