The Higgs field is responsible for the masses of all elementary particles. Including the Higgs particle. But doesn't this transfer the question of mass, or inertia, to the Higgs field itself? Why has the Higgs field inertia which can be transmitted to the massless elementary particles?
The mass of the Higgs is not due to the Higgs mechanism. There simply is a mass-term in the Higgs field Lagrangian.
The whole idea that Higgs is the originator of mass of otherwise massless particles is quite mistaken, at least as it stands in the collective psyche of the popular culture. ;) The correct statement is that coupling to Higgs gives mass to fermions of the Standard Model and they would have been massless otherwise. As such, in the framework of quantum field theory, a field can simply have a mass term -- there is nothing wrong with it. However, when a theory has certain symmetries, it can forbid the field from having the mass because the mass term might violate the symmetry. This is precisely what happens with the fermions of the Standard Model, the $SU(2)_L$ gauge symmetry forbids the mass term. However, due to the coupling of these fermions with the Higgs field, via the mechanism of spontaneous symmetry breaking, they acquire a mass.
In any case, the moral is that there is nothing that says that you need to somehow have a source of mass. The Lagrangian of a field can simply have a mass term. It is the particular case of some field theories where certain particles might not have mass due to symmetries but they nonetheless acquire mass due to spontaneous symmetry breaking. However, this need not be the case. Ironically, as I mentioned earlier, the mass of the Higgs does not come from its coupling with itself. It is massive regardless of spontaneous symmetry breaking.
About mass and inertia.
The two are correlated, obviously, but they're not the same thing.
Example: the mass of each of the elements of the periodic system. The mass of a Helium nucleus is close to two times the mass of a proton plus two times the mass of a neutron, but not quite.
The inertial mass of a nucleus involves the energy state of that nucleus. A higher energy state comes with a correspondingly larger inertial mass.
The inertia of energy is unrelated to the Higgs mechanism.
My understanding is that the Higgs mechanism doesn't even address the question of inertia.
One author phrased this as follows (I'm quoting from memory): coupling to the Higgs field imposes an energy cost. The mass acquired due to coupling to the Higgs field is the inertial mass that corresponds to that energy.