The lottery is an annual tradition. It is a communal event that occurs every year on June 27th. Mr. Summers is one of the two men who administer the lottery. It is perhaps significant that his name is Summers and the ritual sacrifice occurs in the summer. His name is therefore linked to the annual event. His name represents a repeating cycle and a repeating tradition which, to some people like Old Man Warner, is as natural as the summer season itself.
Mr. Graves' name symbolizes death and/or a grave, a place where the dead are buried. The word "grave" can also be used as an adjective and it can mean to be serious, solemn, and threatening. This describes the way the townspeople feel about this barbaric ritual.
Mr. Graves is the postmaster, a civic job. It is also his civic duty to swear Mr. Summers in as the official of the lottery. The man named Graves literally delivers (like the mail) the man who will announce the sacrifice.
When it is Mr. Graves' turn to pick from the box, Jackson emphasizes the symbolism of his name to show how solemn and dark this ritual is:
"We're next," Mrs. Graves said. She watched while Mr. Graves came around from the side of the box, greeted Mr. Summers gravely, and selected a slip of paper from the box.
Each year, the whole town "greets summer gravely" because the lottery occurs in the summer. It is something they dread. Thus, when the lottery occurs, each person greets Mr. Summers gravely because he runs the lottery.