Forgive a humble engineer for interrupting when physicists are speaking about the secrets of the universe, but something about your statement struck me as strange.
I am used to thinking about mass (gravitational or inertial) as being a fundamental property of things, not to be explained so much as described or measured. The concept of mass would seem to be more basic than the Higgs mechanism; how, then, can the Higgs mechanism explain why things have mass?
Would you agree that degrees of freedom are more fundamental than properties? The Higgs mechanism works on a mathematical level by making an extra degree of freedom available to the elementary particles, and this degree of freedom manifests itself as mass. (There are also extra degrees of freedom left over known as Goldstone bosons; the Higgs particle itself is an example of a Goldstone boson.)
The physical interpretation of that math would go like this: the "massless" elementary particles are coupled to the Higgs field, which "dresses" the particles in a cloak of virtual Higgs particles, and it is this cloak that plays the role of mass. The stronger the coupling, the heavier the cloak.
You might want to look at the site, Higgs Revealed. It has several accounts written in response to a challenge to explain the Higgs boson on 1 page.
If mass were basic, then would it be convertible into something else, such as energy? What they do is rotate their dimensional unit matrices until they become relatively simple; then one of the dimension units becomes mass. But there are other possibilities.