Schaff, Philip. "Ch. 29: Authorship." The Oldest Church Manual Called the Teaching of the Twelve Apostles. Second Edition, Revised and Enlarged. New York: Funk & Wagnalls, 1886, 125-127.
The author of the Didache is not identified. Schaff considers this a mark of modesty. He also thinks the author is similar in tone and theology to Matthew and James, rather than Paul (Schaff 1886, 125).
Despite any differences in theology, the author of the Didache refers to the "Twelve Apostles" in a favorable way. He avoids polemics quite consistently, preferring to make a model of respectful and faithful Christianity (Schaff 1886, 126).
Though the authorship remains a myster, Schaff notes two speculations, that of Symeon of Jerusalem, who succeeded James as leader, or possibly to an anonymous author in Antioch reacting to Barnabas after the clash of Peter and Paul (Schaff 1886, 127). This is, however, all speculative.