Update: My answer to this question.
There are lots of properties we give to various particles, like the electron or neutron. A fundamental physical particle has a mass, a spin, perhaps an electric charge, perhaps a color charge etc. etc. You are most familiar with the electric charge from Coulomb's law, that the force between two charged objects is given by
Similarly, there is a force between two massive particles in Newtonian mechanics
From this 'force law' perspective there's nothing fundamentally different between the electric charge on an object and it's mass. In fact, the mass can just be thought of the 'gravitational charge' of an object.
Mass is special though, but not because there is anything special about Newton's Law of Gravitation. Rather, the special role of mass in physics is generated by one of Newton's other laws, namely his Second Law
In principle the m in Newton's second law and the m in Newton's gravitational law could be different as the F=ma mass is the "inertial mass" which plays a different role than the gravitational mass. We therefore have two questions:
Is there a reason the inertial mass is equal to the gravitational mass? (And for anyone who is thinking "because the equivalence principle says so", that's a circular argument so you'll need to go further.)
What would happen if Newton's second law was actually , where q is the electric charge and force was measured in ? In other words, does the fact that place restrictions on what m can be for particles?