Up to this point, the narrator has been nameless (apart from being briefly given the false name, Elly Higginbottom, when she and Doreen spend the evening with Lenny Shepherd).
In biographies, 'Greenwood' is given as the maiden name of Sylvia Plath's maternal grandmother although, as she was born in Austria, 'Greenwood' is presumably the americanisation of a German surname.
The name 'Greenwood'* may also have been chosen to convey a sense of youth, immaturity, and naivety, reflecting the narrator's lack of experience in certain areas of life. 'Greenwood' is also an old English name for a forest.
The name 'Esther' may have specific personal associations for Sylvia Plath, or she may have had in mind the biblical Esther, who was responsible for the deliverance of the Jewish people.
(* 'green wood' is freshly felled wood which is still 'sappy')