My understanding was that the "Gig 'Em" came from their rivalry with TCU back in the day. TCU was declared national champion in both 1935 and 1938 (as well as featuring the 1938 Heisman recipient). Aggy won their one and only legitimate championship in 1939 (shared with USC), so the rivalry was quite strong back in the late 1930s and early 40s. Unfortunately, for tamu, every single one of their students was called away to individually fight to save our way of life, while no other American contributed in any way whatsoever to the war effort. Because tamu was called on to single-handedly save civilization, their football program declined.
This is the time frame that aggys call their "war period" and evidently lasted lasted until the U.S. left Iraq because between 1939 and 2012 the aggys never managed to have a team that could win bowl games in consecutive years.
The ags claim the story is similar, but the ags have so many stories that they worship that are just fabricated lies to make themselves appear to be more than they are, I don't know how much credibility to give the story until someone than an aggy can speak on the matter:
This tradition began at a 1930 Midnight Yell Practice held before the football game against the Texas Christian University Horned Frogs. In an attempt to excite the crowd, Pinky Downs, a 1906 Texas A&M graduate and member of the school's Board of Regents, asked "What are we going to do to those Horned Frogs?" Using a term for frog hunting, he answered his own question, "Gig 'em, Aggies!" For emphasis, he made a fist with the thumb extended. The phrase and hand signal proved popular, and it became the first hand sign of the Southwest Conference. Gig 'em is also the name of one of the school yells, which is used during football kickoffs.
While on the subject of "back in the day," there has been an ongoing thread on the Shag that discusses the various elements of aggy stupidity. One of the ags in that thread popped off about the "greatness" of their little dog and offered the claim that the "greatness" wasn't an aggy fabrication, rather he claimed it was a "greatness" recognized by the U.S. Army in awarding their dog an honorary rank, supposedly as a sign of the thanks of a gratified nation in recognition of the overwhelming sacrifice made by tamu, its students and alumni in saving humanity during WWII.
Here is what I managed to come up with in trying to verify the claim the aggy made:
I did a quick search and found out the entire "The Army gave our mascot the rank of general" story is, of course, total horsesh!t.
During World War II, there was a civilian organization called "Dogs for Defense, Inc." It was headquartered in New York City and it assisted with the nation's collecting pet dogs for training as dogs for military use. Starting in 1943 they had a fund raising drive called the 'War Dog Fund." This totally civilian organization conferred a completely fake rank on anyone's pet for a fee. For a donation, an individual got a tag for the dog, a thank you letter and a certificate attesting to the dog's purchased rank. For $1, the dog became a private in the Army or a yeoman in the Navy. Different ranks were given for different amounts, all the way up to $100 to make your dog either an Army General or a Navy Admiral. Not a U.S. Army General or U.S. Navy Admiral, an "army" general or "navy" admiral in the War Dog Fund.
Some people got together in 1943 and decided to donate $100 in the name of the aggy mascot. Through the years, the ags have made up an entire story stating the dog is not just a general, but a "5-diamond cadet general", that the honor was awarded by the U.S. Army and that it was done as an expression of the nation's appreciation for the overwhelming sacrifice the university made in sending its students to fight the war. All this is (again) typical aggy horsesh!t. The award did not come from any governmental agency, it had nothing to do with the U.S. military, it was available to anyone who wanted to donate the money, it carries no significance, the award wasn't that the dog was a "cadet general" (there was no such "cadet general" rank given out) and it wasn't in appreciation for anything other than the $100 donation. This "rank," much like the aggy corps itself, has nothing to do with the U.S. military and the U.S. military has nothing to do with it.