This is an interesting question. Yes, the Higgs Boson goes some way to explaining inertia. But it leaves open a very big question.
The first thing I should say is we don’t really need the Higgs Boson for most of the mass in the Universe. Almost all of the mass in the Universe comes from the “potential” energy of different particles. If we store particles in a certain state like an atom or quarks in a proton, it has a certain potential energy- the energy taken to “bind” a particle into that particular state. Then, by Einstein’s E=mc^2, we can associate a certain “mass” with that potential energy.
This is why every single complex particle in the Universe- those made from smaller particles has a mass, and most are relatively heavy.
What the Higgs Boson does is give a way for the fundamental particles- those not made from anything smaller- to gain a mass. They pick up this mass from interacting with the Higgs Boson’s field.
The type of mass the Higgs field gives these particles is inertial mass. What the Higgs field does is give them a certain sort of resistance to being accelerated by other forces- in exactly the way mass does in Newton’s equation F=ma.
But the big mystery about mass is why gravitational mass and inertia are the same thing. Inertial mass is the way we normally encounter mass: as a resistance to forces or change, it’s the mass in Newton’s famous equation F=ma.
Gravitational mass is the tendency of mass to attract other masses towards each other using the force of gravity.
Nobody in physics yet understands why they are the same thing.
What we really need to understand mass is a quantum theory of gravity. Nobody has developed a really convincing theory of quantum gravity yet! This field is ripe with opportunities.