Skimming through Minkowski's famous 1907 paper, he uses the term ponderomotive force.
What does he mean by this?
He just means the Lorentz force. The Lorentz force is called the "pondermotive force" in his paper, for no good reason. Old papers did not have internet to standardize their terminology for them.
Let's look at some clues as to what it probably meant at the time. The word is ponderomotive rather than pondermotive and is constructed like electromotive, magnetomotive, from ponder-o-motive. The [etymology] of ponder is given as
Therefore as an initial guess, it could mean the line integral between two points of a force that acts upon substance to give it weight; perhaps the line integral of the Newtonian gravitational force?
Book Googling 'ponderomotive' turns up a quote from Energy and Empire: a biographical study of Lord Kelvin
Hermann von Hermholtz and the foundations of nineteenth-centurey science by David Cahan
Page 11 of Eddington's Principle in the Philosophy of Science
Page 165 of a 1922 Bulletin of the National Research Council By National Research Council (U.S.)
So Minkowski meant the electromagnetic force on mass - the Lorentz force.
Ron's responses are incorrect. You can see here that it was Boot and Harvie in 1957. In an inhomogenous plasma all particles regardless of charge will move toward the weaker field. This is much different from Lorentz forces, ie the motion of neutral particles do not generate a magnetic field.