In The History and Topography of South Crosland, Armitage Bridge and Netherton (1938) by Philip Ahier, the author gives the following suggestion for the derivation of "Mag" in relation to the former hamlet of Mag Lordship which was part of the township of South Crosland:
The prefix of Magdale and Magbrook is probably derived from an old Norse word "magi," which literally means "stomach" or "belly," and is used in this sense to denote a narrow river gorge.
However, a more likely explanation by Dr. George Redmonds is given in Magdale and Steps: Life and Industry in a Textile Hamlet (2008) by Alan J. Brooke:
The name Magdale does not occur until the late 1830s, but the element "Mag" in local place names is found as early as the mid 1400s [...] Dr. Redmonds suggests that both the origin of the name Mag and of the lordship comes from Margaret Beaumont of Crosland Hall who, in the period 1356-71, was involved in an inheritance dispute following her husband John's death. This may have led to a division of the manor and the creation of the lordship consequently known, along with the river marking its boundary, by the diminutive of Margaret, Mag.