Actually the constraint $$ \frac{1}{c^2}\frac{\partial\phi}{\partial t} + \nabla \cdot \mathbf A = 0 $$ is due to Lorenz (Lorenz and Lorentz are easily confused): https://en.wikipedia.org/wiki/Lorenz_gauge_condition This equation is sometimes used because it leads to simple and symmetric wave equations for the scalar and vector potential, which are then easily solved for known charge and current distribution and initial conditions on the field. The potentials are auxiliary functions without direct physical meaning. The meaning of the constraint is really just simplification of the relativistic equations so they become nice and simple.
You confuse me a bit with the speed-of-light factor. In relativistically covariant notation, it's [tex]\partial_{\mu} A^{\mu}=0.[/tex] Split into temporal and spatial components this reads [tex]\partial_0 A^0+\vec{\nabla} \cdot \vec{A}=\frac{1}{c} \partial_t \Phi + \vec{\nabla} \cdot \vec{A}.[/tex] This is, of course, in Heaviside-Lorentz units. The good thing with this particular gauge, which should indeed be named after the Danish physicists Ludvig Lorenz instead of the Dutch physicist Hendrik Antoon Lorentz, because Lorenz was the first, using this gauge condition. The physical merit of this particular gauge is clear: It's a Poincare invariant condition, leading to Poincare invariant equations of motion for the four-potential that at the same time separate into the components. This makes it particularly nice for radiation problems. For other problems like the description of bound states in quantum mechanics other gauges are more convenient. In this case the Coulomb gauge is good. It always depends on the physical problem you want to solve, what's the most appropriate gauge constraint. Choosing a gauge is an art comparable to the one to find the most convenient set of coordinates to solve a problem.