(noun and verb) nudge; moderate run or ride
The word jog appeared in the 16th century. Its etymology is uncertain: it may well be an adaptation of shog, a Germanic word with a similar meaning that had been in English since the late 14th century, and which is a favourite expression of Nym in Henry V (II.i.42, II.iii.42). "Shall we shog?" ("Let's go") has since become something of a catchphrase for bardaholics.
Alternatively, it might have been a fresh onomatopoeic coinage, the sounds of the word reflecting the jerky movements involved.
The basic sense is "move along", especially with the idea of "moving off or away". This is the meaning we need when we hear Autolycus sing: "Jog on, jog on, the footpath way" (The Winter's Tale, IV.iii.121).
The modern sense might well apply here without being at all misleading, of course. But when Katherina tells Petruchio: "You may be jogging whiles your boots are green" (The Taming of the Shrew, III.ii.210), she is telling him to go away, not advising him to take an early-morning gentle run.