Evidence for Brendan McKay's suggestion that it comes from the older concept of an alternating polynomial may be found in Burnside's classic book, *Theory of Groups of Finite Order*. When defining the symmetric and alternating groups, he puts an asterisk next to the word "alternating" and adds the following footnote:

The symmetric group has been so called because the only functions of the $n$ symbols which are unaltered by all the substitutions of the group are the symmetric functions. All the substitutions of the alternating group leave the square root of the discriminant unaltered.

By "the square root of the discriminant," Burnside means the polynomial
$$\prod_{r=1}^{n-1}\prod_{s=r+1}^n (a_r-a_s),$$
which of course is an alternating polynomial.