Chavah [HWH] is named by Adam [Breisheet 3:20] after the incident with the Nahash. Until then she is merely "ha'Ishah"
R.S.R. Hirsch points out an interesting word play with the Aramaic word "Chivya" [Hiwyah].
The root of the Hebrew Chavah and the Aramaic Chivyah [snake] are virtually the same. This suggests a very different approach to Chavah's name than the text's own reason [viz. Eim kol Ha]. At any rate, both reasons are apparently the result of the Nahash incidedent
See R.S.R. Hirsch Breisheet 3:6 p. 99 s.v. Sa'avah in the New Edition.