What is the origin of the term "Gee Gee" when referring to racehorses?

The term Gee Gee came about when Children in the early 19th century saw horses on a daily basis, in many cases. The youngest children, just learning to speak, would hear men shouting "Gee!" to their horses, and so they, very logically for children, applied that word to the animal. It became gee-gee after the pattern of other children's words for animals, such as bow-wow for dog and kitty cat for cat, though it was still found as gee alone, as well.

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When working with horses, the command "Gee" is used both as an abbreviation of "Gee-up" to mean go forward, and as an instruction to turn right. ("Haw" is the instruction to turn left.) In a time when horses were a part of daily life, you would have heard the term "Gee" addressed to them regularly; hence the transfer from an instruction to an epithet described by merln31.