According to Wikipedia, the name is not historical (i.e. he didn't use it while he lived, nor did his contemporaries use it of him), and may have been based on Thomas à Kempis. read more
Thomas Becket was not called Thomas à Becket during his own lifetime. He was called Thomas à Becket, by people who wrote about him much later, possibly in imitation of the name Thomas à Kempis, who was another medieval monk. read more
St. Thomas Becket, also called Thomas à Becket or Thomas of London, (born c. 1118, Cheapside, London, England—died December 29, 1170, Canterbury, Kent; canonized 1173; feast day December 29), chancellor of England (1155–62) and archbishop of Canterbury (1162–70) during the reign of King Henry II. read more