What is energy in quantum mechanics? Is it wrong to say energy is the expectation value of Hamiltonian?
Or should I say energy is the eigenvalue of Hamiltonian?
 A: You must be a bit more explicit in your language than in the classical case. Either could be correct but I lean towards the eigenvalue description and I'll explain why.
First of all, the Hamiltonian $\hat{H}$ is something which "belongs" to system. Energy is something which belongs to a state. So for example if a state $|\psi_n\rangle$ has
$$
\hat{H}|\psi_n\rangle = E_n |\psi_n\rangle
$$
We can unambiguously say "this state has energy $E_n$" because every time you measure it you will get energy $E_n$ and also the average energy of this state, $E_{avg}$ is $E_n$.
However, as was just pointed out in Superposition principle forbids quantisation? we can consider states which are superpositions of energy eigenstates such as $a|\psi_n\rangle + b|\psi_m\rangle$ which can have average energy anywhere between $E_n$ and $E_m$. One could say it is a state with this new energy, $E_{avg} = |a|^2E_n + |b|^2E_m$ but I think that would be misleading because if you do a measurement on this state you will never measure energy $E_{avg}$, you will always get either $E_n$ or $E_m$. 
The language I would use is either that it is a state which is in a superposition of energy states or (more daringly) it is a state which both has energy $E_n$ and $E_m$.
