Where the word "Hanto" 半東 came from is difficult to discover. The scholarly Daimyo Tea Master, Matsudaira Fumai had to make do with the explanation that it was a vulgarism of Chinese origin. Indeed even Tsutsui Hiroichi came up with the far fetched idea that the Zen word for the person in charge of the rice and gruel in the monastic kitchen, properly known as the "hanshuu" 飯頭 which nowadays can be pronouned "hantou," was metamorphed by Fumai's time into the present hantou, based on the Hanto being only "half the east" 半東. Another explanation is that in the old days the Teishu used to sit to the east and the guests used to sit to the west, so the direction east came to symbolize the master. "Half the east" (Hanto) therefore came to mean assistant to the master. This makes sense since a properly aligned tea room has tokonoma (alcove) in the north, and throughout this period the dominant form of tea room was Gyakugatte, where the master sits to the east and the guests to the west. Interestingly enough, in the beginning the word "Hanto" was regarded as informal jargon, and some people thought that it should rather be called "Han Teishu."
As to the origins of the position Hanto, I have heard two explanations. One was that the last guest at a chaji was the representative of the plantation from which the Tea master's tea had come, thus his or her title "Tsume", and he helped the Tea master grind the koicha and usucha. From that, the idea of helping do everything grew naturally. Having spent the first half of the chaji grinding the tea, the tsume only joined at go-iri (and would also die if the tea were poisoned).
The other explanation is that the Hanto was the Tsume, last guest but was a relative or disciple of the Teishu and thus knew the tearoom and house intimately. Thus the Tsume is the only one able to go into the mizuya, if necessary. We see the Tsume used as the Hanto especially during the kaiseki and in bringing things to the Shokyaku and sadoguchi. Again, the idea of someone else helping do everything grew naturally.