Dr. Watson's first name is John. His full name is John H. Watson. The H stands for Hamish. Dr. Watson's name is only mentioned four times in the original works by Sir Arthur Conan Doyle.
Usually, the name is listed as a credit to Dr. Watson for his contributions to the story rather than in the story itself. Dr. Watson himself does mention his own name, however, in "The Problem of Thor Bridge." While his wife, Mary, calls him James in certain texts, it is believed that she may be referring to his middle name, Hamish, which is a variation of James.