The relativistic dispersion relation states:
\begin{equation} \frac{E^{2}}{c^{2}}=m_{0}^{2} c^{2}+p^{2} \end{equation}
Where $m_{0}$ is the rest mass.
I have heard that the mass ($m_{0}$ in this context) can be defined as the gap of the dispersion relation.
Take $c=1$ and draw the $E$-vs-$p$ hyperbola. The gap is the gap between it and the origin (or between it and the line $E=0$); that distance in the energy-momentum plane is equal to $m_0$.
The physical significance of this gap is that a massive particle has a minimum nonzero energy, even when it has no momentum.
In my opinion, it’s better to think of the mass as the Lorentz-invariant length, in 4D Minkowski space, of the energy-momentum four-vector $(E,p_x,p_y,p_z)$, namely
$$m_0=\sqrt{E^2-p_x^2-p_y^2-p_z^2},$$
in units where $c=1$. All inertial observers, with various relative velocities, agree on the value of this 4D length, even though they do not agree on the values of the components $E$, $p_x$, $p_y$, and $p_z$. Many quantities in Relativity are relative (i.e., frame-dependent), but some are absolute (frame-independent), and mass is one if them.
