jog (v.) Look up jog at
1540s, "to shake up and down," perhaps altered from Middle English shoggen "to shake, jolt, move with a jerk" (late 14c.), a word of uncertain origin. Meanings "touch or push slightly," "stir up or stimulate by hint or push," and "walk or ride with a jolting pace" all are from 16c.

The modern sense in reference to running as training mostly dates from 1948; at first a regimen for athletes, it became a popular fad c. 1967. Perhaps this sense is extended from its use in horsemanship.
Jogging. The act of exercising, or working a horse to keep him in condition, or to prepare him for a race. There is no development in jogging, and it is wholly a preliminary exercise to bring the muscular organization to the point of sustained, determined action. [Samuel L. Boardman, "Handbook of the Turf," New York, 1910]
Related: Jogged; jogging.
jog (n.) Look up jog at
c. 1600, "act of moving up and down," from jog (v.). Meaning "a slight push or nudge" is from 1630s.