Here is a very apt quote from Wikipedia:

…the combination of spontaneous symmetry breaking in conjunction with the Higgs mechanism gives these particles mass. It also suggests the presence of a new, as yet undetected particle, the Higgs boson.

This makes clear something that I think is glossed over all too often: the Higgs boson is a consequence, a side effect, of the proposed theoretical description of why particles have mass. The Higgs particle itself does not actually bestow mass on other particles.