According to http://www.word-detective.com/0806A.html , 'many small children had noticed that a lot of grownups were shouting "gee" at horses and decided that "gee" was another name for "horse."' but according to http://en.wikipedia.org/wiki/Ottawa_Gee-Gees , 'The term "Gee-Gee" is taken from horse racing where a Gee-Gee is the first horse out of the starting gate.' Are these correct?
This Wikipedia article gives this explanation for the origin of the word gee-gee:
The Chester Racecourse site was home to the famous and bloody Goteddsday football match. The game was very violent and, in 1533, banned by the city, to be replaced in 1539 by horse racing. The first recorded race was held on February 9, 1539 with the consent of the Mayor Henry Gee, whose name led to the use of the term "gee-gee" for horses.
 a b Marcus Armytage (2008-05-06). "Chester racecourse moves with the times". The Telegraph. Retrieved 2008-05-09.
The founding father of Chester Races, Henry Gee, was a zealous reformer who is said to have put the corporation house in order "With a high hand and an unswerving purpose". He suppressed corrupt municipal practices and appointments (where is he now, we wonder?), banished "Idle beggars and vagabonds", regulated the markets and established the first attempt at a school board. He banned single women from keeping common ale-houses and "Stamped on immorality wherever he found it". Henry Gee died in 1545, but his name is remembered in the running of the Henry Gee Stakes for three-year-old maidens at the July meeting, and possibly also in the old, but still commonly-used, English nickname for racehorses: Gee-Gees.
This definition of gee-gee says it is a UK child's word for horse. I don't believe that is used by children in the U.S.
I don't know if this is relevant, but gee or gee up is also a command to get a horse to move faster. (It is also used to have a draft animal turn right. As opposed to haw, which is a command to turn to the left.)
From "Horse racing jokes" on http://www.ukjockey.com/jokes.html
How do you spell 'Hungry Horse' in four letters?
M T G G (Empty Gee-Gee).
Irish farmers, including my ancestors, settled in the Ottawa Valley, Canada, in the 19th century. They brought the term Gee-Gee with them from the British Isles. They used it to refer to their big farm horses, as well as race horses. The colours of the University of Ottawa are Garnet and Grey, or GG for short. For a long time the uOttawa sports teams were known as the GG's. Over time this morphed into Gee-Gees, the same as the horses of the Ottawa valley. In 1970 uOttawa adopted, as their official sports team logo, the head of racehorse with a fiery mane. The letters GG are emblazoned, in grey, in the center of the garnet horse's head. They can be taken to represent both the team colours and the team name, Gee-Gee.