I have read that the eigenfunctions of any hermitian operator can be used as a basis to express any function, but I have only ever really seen the eigenfunctions of the Hamiltonian used. Why is this? Could I express the hydrogen electron in terms of momentum eigenfunctions?
General wave functions can be expressed in terms of any set of eigenfunctions.
But for bound systems, the energy eigenfunctions have a couple of appealing properties that make them popular:
The energy eigenfunctions are the solutions to the time-independent problem, so you can work on a steady-state system. This often makes the math a lot easier.
The energy eigenvalues may be easily accessible in the lab. So, after making some necessary assumptions and approximation to get through the math you can check that your results agree well with reality. And spectroscopy is an incredibly fine-grained tool, so you can check that your detailed results agree with reality as you relax one approximation after another and approach an analytic or numeric solution to the full physics of the situation.
I don't have any particular thoughts about the use of energy eigenfunction over any other set for unbound systems, however.