Alexandra Honeywell is a beautiful woman with "long, straight hair," that "was indeed the color of honey," portrayed on the news in association with a commune called "Starchild Colony." Edgar sees her, irregularly, in news clips on TV, inviting people to her commune, advertizing it as a "place of peace." He thinks she is beautiful.
Despite her seemingly important introduction toward the beginning of the story, Alexandra Honeywell remains an undeveloped character. In fact, Edgar never even meets her. However, in Part 4, when he is running away and does not know where he is running, the idea of Starchild Colony (and Alexandra Honeywell) compel him to move forward. Again, he does not reach Starchild Colony, nor does he meet the object of his fantasy. The author himself notes however, in a book club Q&A on Oprah.com, that it is the presence of Starchild Colony and the face of Alexandra Honeywell that bring Edgar to the realization that he truly only belongs in one place, and that is home.
As a character, Alexandra Honeywell is unimportant, but as an idea, she embodies the catalyst that sparks Edgar into action.