According to sources quoted in Wikipedia's sinc page, Woodward introduced both the name and the abbreviation in 1952.

According to sources quoted in Wikipedia's sinc "talk" page,

The name sinus cardinalis dates back by Edmund T. Whittaker in 1915, where he named the bandlimited or most simple function of a family of cotabular functions (sharing a function table, i.e., values at equally spaced sample points) the cardinal function of this family. One would have to check if he already named the basis functions or if that came later. See A History of interpolation.--LutzL (talk)

Following that "talk" link leads to a history page which includes an image from that paper, From E. T. Whittaker, "On the Functions which are Represented by the Expansions of Interpolation-Theory", Proceedings of the Royal Society of Edinburgh, vol. 35, 1915, pp. 181-194