The terminology varies by context, but what we're basically talking about is the specific heat capacity of the material. Or rather, if the battery (or other object) were of a uniform material, that would be the specific heat capacity of the material. Of course, it's not a uniform material, so we're talking about the total heat capacity of the sum of components divided by the total mass.
The term "thermal mass" kind of makes sense, as it's mass in which thermal energy is stored, but the language is a bit at odds with other analgous systems. In a simple thermal model, thermal resistance and heat capacity are analgous to friction and strain in a mechanical system or resistance and capacitance in an electrical system. More specifically, this heat capacity is associated with potential energy storage in the system. Contrast this with the energy associated with flows -- moving mass in mechanical systems or inductance in electrical systems. In this sense, calling it thermal mass might be a bit confusing.
So long story short, thermal mass or other measures of specific heat capacity describe how much heat energy a given mass can store at a given temperature. An object with a larger thermal mass can absorb a lot of heat with a smaller temperature rise -- given that the heat loss path has the same resistance.
This may be of importance when you can't improve the heat loss path significantly. In this case, a component with a large thermal mass can better absorb brief pulses of heat generated by internal dissipation. This is why wirewound resistors typically have better pulse-load ratings than metal film resistors. Their external geometry may mean that they have the same thermal resistance to ambient, but the wirewound resistor can store more energy before the element fuses.
