One should separate the question into two parts, the first of which is philosophical, and the second physics. The philosophical question is resolved by understanding that there are "constants" which are just those that set the system of units, and these are constant for the simple reason that they define our conventional units.
The unit-defining constants philosophically cannot change. They can only be determined relative to physical measurements using physical atoms and light, and these measurements serve to fix our units. The constants which are philosophically incapable of changing are listed below:
- The speed of light c, which defines the unit of space given the unit of time.
- Planck's constant, $\hbar$, which defines the unit of mass-energy in terms of the unit of inverse time.
- Newton's constant, which defines the unit of mass-energy in terms of the unit of space (and in conjunction with the other two, fixes a unique unit of mass, length, and time, the Planck units)
- Boltzmann's constant, which defines the Kelvin in terms of the Joule.
- electromagnetic constants, which define the unit of charge
In terms of Plack units, all physical constants are dimensionless. These are the quantities which are philosophically capable of changing (see this question: units and nature )
So the gravitational constant simply cannot change. It is philosophically meaningless to say that it does change. What you would really be saying is that atoms are changing size relative to Planck units.
Here are some constants that can, in principle, change:
- The charge of the electron in Planck charges (the square of this is called the fine structure constant).
- The mass of the proton in Planck masses (this is more or less the exponential of the strong coupling at the Planck scale)
- The Higgs VEV: this is one unnaturally small parameter in Planck units.
- The cosmological consntant: this is the other unnaturally small parameter.
The other dimensionless constants are rougly of the expected size. The electron-Higgs coupling is a bit small, so the electron is somewhat light compared to other lepton and quark masses, but to 1 part in a thousand, not one part in a billion, so it could still be a coincidence.
Within string theory, all of these dimensional constants are quantities which can change, they are all associated with a particle which represents fluctuations in these quantities. These particles are determined by the geometry of the microscopic space-time. The constants which are constant are those whose low-energy dynamics fixes their value, so that small fluctuations return to where they started, and any change in their value requires energies of order the Planck energy.
At low energies, or outside of string theory, the principle that fixes the charges and masses of the particles is renormalizability considerations. So that the reason the electron charge does not vary is that if it changes from place to place, it is a field, and no field can couple in a renormalizable way to the photon and electron-positron field. They are already dimension 4.
The principle of renormalizability tells you that the only constants you expect to see in a quantum field theory which are natural are the dimensionless coefficients of dimension 4 interactions, like the electron charge, or macroscopic scales determined by logarithmic running, like the mass of the proton. The Higgs VEV is unnatural for this reason, it is a fine-tuned mass scale, and this suggests that there is something left that we don't understand about the Higgs mechanism, which will be sorted out once we have experimental data about the Higgs boson.
The principle of renormalizability is only applicable in a scaling regime where all the energies are much lower than the Planck energy. In this regime, you also expect either Newton's constant to be truly constant, which is Einstein's gravity, or for there to be an extra massless scalar field interacting gravitationally, which is Brans Dicke theory. All other corrections are less renormalization relevant, and scale away at low energies (although Einstein gravity itself is not renormalizable, it is the leading surviving scaling term at low energy, so the renormalizability principle still works). Experimentally, we know that Brans-Dicke fields cannot be working at solar-system scales.
Because of the philosophical freedom of choosing units, Brans and Dicke chose to express their theory in terms of the gravitational constant changing from place to place. This terminology is unfortunate. They could have just as easily framed it as the speed of light changing from place to place, and had the exact same theory. It is best to have G and c both constant, and consider their field as a new scalar field that varies from place to place, with no relation to the unit-defining constants.