Phonons are obtaied by non-relativistic quantization of the lattice vibration. The dispersion relation is given by $\omega=c_s k$ where $c_s$ is the velocity of sound. What can we say about the mass of the phonon? I think it is not possible to compare this relation with the relativistic dispersion relation $E^2=p^2c^2+m^2c^4$ and conclude $m=0$. By mass, I do not mean the effective mass but the rest mass. Certainly, if the rest mass of phonon were zero it would have travelled with the velocity of light in vacuum.
I think in the non-relativistic approximation of the Einstein's energy momentum relation, the same $m$ appears in the non-relativistic kinetic energy $\frac{p^2}{2m}$. Therefore, we can still talk about rest mass in non-relativistic physics.
Moreover, phonon being a goldstone boson should have zero rest mass.
Edit: How does one define the rest mass of the phonon?