For a single particle, the relationship between energy, mass and momentum isPut it another way: if you increase the kinetic energies of the component particles by increasing the speed of the "buzzing around," it increases the total energy, but not the total momentum, because the increases in the individual particles' momenta are in different directions so they cancel out in the sum. Therefore the system mass increases.
On the other hand, if you increase the kinetic energies of the component particles by making the whole system move in one direction, it increases both the total energy and the total momentum, and the two effects cancel out as far as the system mass is concerned.