Origin of mass has puzzled scientists for long time. After the discovery of Higgs Boson, many physicists believe that particles acquire mass as they pass through the Higgs field. In my following article I have proposed that after the big bang, as the particle velocity drop and the matter cool, the kinetic energy of the particle gets converted into the mass energy. This process is reversible. Higgs boson also acquire mass in the same way. As far as I know, Magnetic monopoles may not have anything to do with it.

Article Periodic quantum gravity and cosmology

Interesting question.

Intrinsic mass would no doubt relate to the resistance of a massive body or particle to being moved.In other words, intrinsic mass would amount to omnidirectional inertia, that is, resistance of this body or particle to being displaced whatever direction the cause of displacement is being applied. 

In practical terms, this would mean that a measurable amount of translational kinetic energy needs to be applied from any direction to cause it to change its state of motion.

I am at loss however as to what you mean by "physical" as being opposed to intrinsic.

In mass space interaction, gravity is an effect to observe the observer to  understand the concept of time. Mass is intrinsic and it has a property of space occupying, you know faster is shorter this is an effect observed by an observer for relatively speaking; that it  means space contraction. Mass is invariant quantity but not gravity. Universe is accelerating and expanding : spacial expansion ----> slowing down parameter observed. Time is an another physical quantity.

According to author John David Jackson, Physicist, in Classical Electrodynamics - page number 384,Thus we see that the covariant form ( 11.129 ) of the Lorentz force equation gives the rate of change of  mechanical momentum per unit volume as its space part, and the rate of change of mechanical energy per unit volume as its time part.

According to Herbert Goldstein, Classical Mechanics, 3rd Edition - page number 278, To satisfy the two postulates, the space and time of the special theory consist of a single entity that we refer to as spacetime. This spacetime is the geometric frame work within which we perform physics. In page number - 324, 7.11 The special theory uses ideal inertial frames that are assumed to exist over all of spacetime. In page number - 325, the laws of physics can be expressed in terms of tensors since tensors are geometric objects defined independent of any coordinate system. 

Thanks for clarifying your view on what you mean by "physical".

Note that in my own copy of Jackson, 11.129 lives on page 555.

From the special relativity viewpoint, which you describe, I surmise that the state of motion of a body (massive) would not need any external directed translational kinetic energy contribution to initiate a change in its state of motion as I assume, only that the spacetime curvature "pits" created by other masses would inertly let it be submitted to this change and that it would be this change that would be the cause of the induction of the added kinetic energy involved instead of the latter being the cause of the change.

If I understand this correctly, what would then be your definition of the "intrinsic" properties of mass in context?

I also wonder what makes you possibly suspect that magnetic monopoles could have any relation to mass.

I agree with dr. Michaud that this question is important. To me the concept of mass seems to be as elusive as its origin. No doubt, there is a strong link with inertia via Newtonian mechanics. But let me elaborate a little bit on the inertial origin. Sometimes Newtons law F=ma is interpreted rather as a dynamical definition of mass. In fact the mass as measured with a dynamometer originates from statics. Thus, one could ask what a characterization through movement can tell about mass as such.

Mass has been linked also to chemistry by Lavoisier, who eventually introduced the practice of weighting in chemistry. Again, gravity seems to play a role in the determination of mass. However, Archimedes' principle indicates that the upward force, on which the decision about the quantity of mass is taken, rests on the difference between weight and buoyant force, and mass is not measured as such. Thus, the question may arise if to mass can be attributed a meaning resting on some basic, easily repeatable experiments, or if it is better to try and give a workable definition of it.