Ok, so, a little research and it seems that there is no clear answer as to the origin of Jeep.
Theory 1: Invented by the author of Popeye, who with Olive Oyl, also invented a character named Eugene who is half dog and half cat with a bulbous nose that could magically "go anywhere" and the only words muttered were, "Jeep!" That could fit if one stretched...mentioned in Merriam Webster's dictionary.
Theory 2: (mentioned above) GP is phonetically similar to Jeep, and means "general purpose, "so that is where it came from. However, the G in military designation referred to "Government" and the P was a designation of a recon vehicle with less than 80" wheelbase. Good try...
Theory 3: (mentioned by OP's wife)
. Stands for Just Every Essential Part, referring to the rudimentary nature of these vehicles. True, but most unlikely theory so far...
Theory 4: Jeep was a term used of untested equipment and soldiers before the first Jeep was ever built. Somehow this designation stuck with the Jeep when it was first introduced as untested equipment. Better than the others...
At any rate, the term jeep was already in use before the 1940's and no one really knows how it got attached to our rigs.
https://www.cjponyparts.com/resources/jeep-name-origins
Merriam-Webster, Inc.
Merriam-Webster’s Collegiate Dictionary. Springfield, MA: Merriam-Webster, Inc., 2003.