It is very desirable for $\psi$ to be an eigenfunction of as many operators as possible
In fact, we would probably like it to be an eigenfunction of all angular momenta, but they do not commute, so we can't make it be an eigenfunction of all momenta, so we choose only one (usually $L_z$)
There are many reasons, but the main ones are:
- They are measurable
The Hamiltonian is the energy operator (roughly speaking). If a function is an eigenstate of the Hamiltonian, that means that $\phi_n$ has a defined energy. Energy is a quantity which is easily measurable, so we choose the Hamiltonian to be an important operator in our complete system of commuting operators (CSCO)
In the same way, angular momenta are easily measurable using magnetic fields.
- The fact that a function is an eigenstate of an operator allows us to "label" it with quantum numbers. We can set a state to be $|n\ l\ m_l\ s \ m_s\rangle$ because of that. You cannot have a quantum number if the function is not an eigenstate, because it wouldn't have an eigenvalue to label it with.
So having "something easily measurable to label with" is a good idea. Saying $n=1$ is good because it is easily measurable.
Plus, it is a really basic concept in physics, and we have a lot of intuition on it. If we say that the energy level is the first level, we quickly get an idea of how the electron is. This would not happen with weirder magnitudes.
- The Hamiltonian is more than that
The Hamiltonian, besides the energy, is also responsible of the time evoluition. As it is involved in the Schrödinger's equation, the Hamiltonian rules the time evolution. If something commutes with the Hamiltonian, then that quantity is conserved over time. So if $[H, L_z]=O$, then the value you measure for $L_z$ is conserved over time. So a state with $m_s=+1$, for example, will keep that value over time. That's why the Hamiltonian is important.
The angular momenta are important as well, because things that commute with the angular momentum are invariant under rotations, which is useful, because it tells us if things are symmetric or not. You know that symmetry plays an important role when simplifying problems.
- Commutation itself
The very fact that two operators commute is probably always a good thing. Commutation means that you can measure $H$ and then $L_z$ and vice-versa, and the result is the same. Translation: if you measure energy, it doesn't disrupt the system and you can measure $L_z$ afterwards. If they do not commute, measuring one changes the state for the next measurement.
So, you want them to commute, because you want to measure both things of the same state. It happens to be that "knowing" $L_z$ is not enough to determine the state, there are many states with the same value of $m_l$. We need more information.
However, given a certain energy, with a certain momentum, and certain spin, and so on, the state is unambiguously determined.
That's why we need a complete set of operators. And they must commute so that we can measure all of them without disrupting the system under measurement. That's why we seek for a CSCO.
And, of course, if you need 5 operators to determine your system, you'd better choose operators whose magitudes are easily measurable and have a good physical meaning.
And there are probably more reasons, but I can't remember them all now. Feel free to complete my answer in comments.