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That's not a very useful concept, because it's nothing but the total energy measured in different units. It only has a name of its own for hysterical raisins. A much more useful concept is the invariant mass, which is the square root of the total energy squared minus the total momentum squared (in suitable units), which (as the name suggests) is the same in all frames of references; in particular, it equals the total energy in the frame of reference where the total momentum is zero. Nowadays when people say "mass" they usually mean the invariant mass, because it makes more sense to call the relativistic mass "total energy" instead.