Layman trying to wrap my head around this: the Higgs field causes other fields to stiffen by giving them a resonant frequency, with higher frequencies meaning more mass.
Hmm, now this is making me think, does the Higgs field act like an additional degree of freedom for energy to be dumped into? I mean like a photon is massless, so any amount of energy, it will already be going the speed of light so the only place where additional energy to go into is the frequency. Perhaps with massive particles, a portion of this additional energy now gets dumped into this resonant frequency rather than translating into motion? So the energy stored in this resonant frequency would be like the kinetic energy...? or maybe totally wrong :)
It's potential energy (m_0 c^2), but in a way it's also kinetic because it is a moving wave, it's just that it's a standing wave so it's as though it's reflecting, but being a standing wave causes that part of the particle's bundle of energy to manifest as potential energy.