It was originally called "sanbon toh," which literally means "three"-"tray"-"sugar". It is not clear where the name "sanbon" came from. One story says that the name originates from the shipping port for wasanbon toh, Sanbonmatsu in Kagawa prefecture. Another story is that it came from the Chinese official ranked third in the government who was dealing with wasanbon toh at the port.
The most plausible explanation for the name is that the sugar is refined three(san) times on a tray(bon). When the production of wasanbon toh began for the first time, its refining technique spread. However, people had no special refining boards and they used trays instead. Also, when there was no white sugar, the sugar which went through the "togi" process three times was considered to be refined sugar for shipping.
Imported pure white sugar was often called "sanbon-siro("siro" means white in Japanese)" when sold. In order to ensure that the sanbon toh is domestic sugar, "Wa", which means Japanese, was added. Thus the name "wasanbon toh" came into being.
A little wasanbon toh is made in Kagawa prefecture. In order to emphasize it being a product of Tokushima prefecture, the old name for Tokushima, "Awa" was added to "wasanbon toh". Totally that's name became "Awa Wasanbon Toh". Incidentally in Kagawa the same sugar is usually called "Sanuki sanbon" or "Sanuki sanpaku".