There are several interesting possibilities for an essay on "Young Goodman Brown," a short story that has intrigued literary critics since its publication.
A standard, but interesting, subject for analysis is Hawthorne's use of symbols in the story. For example, Goodman Brown's wife, Faith, and her pink ribbons are very interesting subjects to analyze in the context of Brown's own faith. Other symbols that yield interesting essays are such elements as the forest and Native Americans, which represented very specific things to the colonists.
Students have written very successful essays analyzing the role of Satan in the story--and some students analyze the devil as an real entity who interacts with YGB or as a figment of YGB's imagination, based on YGB's conception of what Satan should be like.
Another subject you might want to consider is whether or not Young Goodman Brown actually experienced any of the events he recounts in the story or whether all the experiences were part of a dream.
A more complicated, but very worthwhile, subject for an essay is an analysis of the nature of the Puritan religion itself, particularly in its conception of the devil's actual, physical presence in the community. Another topic is an analysis of the Puritan religion and its effects, whether positive or negative, on the psyche of an individual like Young Goodman Brown.