As with the development of all papers and writing samples much will be present with what you have in front of you. Simply put, determining your thesis statement might be contingent on what you know and what you can argue. There are many elements to Jackson's work which you can analyze or study. I think that being able to assess which component of it you will spend time addressing in your writing sample and what ideas or concepts you can explain through writing will be another part of this process. Once these areas have been settled, I think you will find it easier to construct your thesis statement. Until then, it will be a bit more difficult. Simplicity might be the first step in this process. What did you get out of the work? What is it you are being asked to do in analyzing the work in your writing? These steps along with assembling what is in front of you should help in constructing a thesis statement and then a paper.
In Understanding Fiction, Cleanth Brooks and Robert Penn Warren suggest unease with the story's structure, contending that Shirley Jackson has
preferred to give no key to her parable, but to leave its meaning to our inference.
Working with "The Lottery" as a parable, why not focus on the symbols as keys to this parable? One approach sees certain characters as representative of social classes: Mr. Graves and Mr. Summer represent the upper class because of their power and money, while Bill Hutchinson and Mr. Adams represent the working class. The lottery represents fate that determines people's social class.
Another approach has the names suggestive of other things. Mr. Graves represents the imminent death that waits for the person with the black mark. Even Mrs. Delacroix--the French word choix means cross--has symbolic meaning, as does Mr. Summers, whose name represents the time of year of the lottery. The black box is the coffin/doom and forthcoming death; the lottery represents the random and senseless acts of violence in human beings.
So, a thesis about symbols could be about how they give meaning to Jackson's parable about man in her short story, "The Lottery."
See the site below for additional criticisms:
A strong thesis will answer a leading question or more than one leading question. So some questions you might want to ask yourself that might help are:
What are the major symbols Jackson uses?
Why do you believe she uses them, and what do these symbols say about the villagers themselves?
What do the villagers themselves symbolize?
I have attached the reference to the enotes page.