The significance of Mr. Summers’ name is that it highlights his central role in the lottery and the life of the village as a whole, as well as being deeply ironic.
Mr. Summers is the most important of all those who manage the lottery, which is held on June 27th, “a full-summer day.” He runs most of the things that the town does collectively, because he has “time and energy to devote to civic activities.” We see here a link between the name and the season it evokes, the summertime when people are most active and presumably happiest. He is the conductor of the lottery, the one who arranges it and declares it “open,” and the custodian of its implements when they are being prepared:
The night before the lottery, Mr. Summers and Mr. Graves made up the slips of paper and put them in the box, and it was then taken to the safe of Mr. Summers’ coal company until Mr. Summers was ready to take it to the square next morning.
He is also the one who remarks significantly on Tessie’s near-absence from the ceremonies (“Thought we were going to have to get on without you”), and finally pronounces what amounts to the formal sentence of execution on Tessie when she proves to be the choice of the lottery for that year:
“It’s Tessie,” Mr. Summers said, and his voice was hushed. “Show us her paper, Bill.”
“All right, folks,” Mr. Summers said. “Let’s finish quickly.”
Mr. Summers is thus the central person in the lottery and the village as a whole, its “sun” so to speak, in full control on this "full-summer" day. But his name is also ironic, in that this “summer” brings with it not only light and warmth, but also, for one character, darkness and death.