With all due respect to Peter Higgs, Steven Weinberg and all other contributors to the Standard Model (SM), I don't think that Higgs' mechanism explains the mass origin.
In my opinion, the main trouble is not that the shape of the Higgs field potential in the SM is identified on ad hoc basis, and the value of the Higgs boson's mass by no means can be identified from the theory.
What I really don't like is that Higgs boson can distinguish, say, between electron and muon fields, and interact with them with different strengths. Electron and muon fields in the SM are both massless and identical. How does Higgs boson know what is electron and what is muon?
Article Electromagnetic Mass, Charge and Spin