Is Motion One Of The Three Eternal Properties of Atoms? I.E. Are The Three Properties Shape, Size, and MOTION?

Here is my comment to Patrikios' thought experiment:
Placing the object outside Earth’s gravity would not be enough because then it could be in the Sun's gravity. Placing it outside the Sun’s gravity would still not be enough because then it could be in the gravity of the Milky way.
We might find a point between two galaxies at which the gravitational forces of both galaxies cancel out each other and where the gravity of further away galaxies is too small to be significant for a reasonably long time of observation.
We can define a frame of reference there at which the object would stay at rest once placed there without initial speed in that frame of reference. (There is a logical circle in it because the definition of an inertial frame of reference is that an object without external force keeps moving with its initial speed, i.e. stays at its place if initially at rest.)
If we want to move that object, we need to apply a force to accelerate it. The proportionality constant between force and acceleration is the inertial mass. (I would not call that "potential for movement".)
If we place the object in a gravitational field, the gravitational mass of the object is the proportionality constant between the force on the object and the gravitational field strength.
Gravitational mass and inertial mass are conceptionally different. We have defined our system of measurement units such that both masses are measured in the same units.
In a system, in which the units are not the same, the two types of masses are proportional to each other but not identical, which makes the conceptional difference more obvious. This is an ongoing topic of research into the foreseeable future. Physicists keep designing new experiments to test the proportionality between gravitational mass and inertial mass with ever increasing measurement accuracy.

In a Newtonian universe with absolute space and time and when there is a point in time where there is no motion of that atom in the chosen frame of reference, yes, this would be the case at all other points in time, the atom would never move and would have never moved in the past.
In an Einsteinian universe with only one atom, there would be no references to measure space and time. Therefore, statements like "the atom moves" or "the atom does not move" would have no meaning.

The case of two atoms in an infinite void:
In a Newtonian universe with absolute space and time and when there is a point in time where there is no motion of both atoms and there is a distance between them, both atoms would accelerate from that point in time toward each other, collide, move with decreasing speed back to their points of rest, simultaneously come to rest there and accelerate again toward each other, repeating that cycle all over again into eternity and would have done so in all eternity of the past. (This is the simplest example of an eternal, pulsating universe. Until the discovery of the accelerated expansion, eternal pulsation between big bang and maximum extension was a credible scenario for our universe.)
In an Einsteinian universe, the two atoms would behave similar to the two atoms in a Newtonian universe. The distance between resting points would provide a reference for measuring space in the direction along which the atoms move. The period for one pulsation would provide a reference for measuring time.

There is a caveat: The physics of a universe might depend on what is in it. So, the laws which we know for our universe of maybe 10^89 particles might not apply in a universe with considerably less particles.