This was not the case of a "split state". There was a "Province of Carolina", created 1629 and governed by a group of eight wealthy English noblemen, and their heirs. The province was divided into North and South a century later, due to disagreements about governing the terrotory. The separation became final after the noblemen sold their interests to the English monarch, and both North Carolina and South Carolina became royal colonies. This was still the province of Carolina, divided for ease of administration. By the time they became "states", the names were already established and there was no major incentive to change it. Similary, the Dakotas used to be "Dakota Territory", named after the groups of natives there. The southern part was at some point already considered separate, with the United States purchasing the territory from the French. It was admited as two states, again, for ease of administration.

Source: Encyclopedia Britannica.