Shirley Jackson (in her short story "The Lottery") makes brilliant use of symbolism with regard to the names of characters in the story, as well as symbols of the black box, the black dot, the stones, etc.

Symbolism is defined as...

...when an object is meant to be representative of something or an idea greater than the object itself.

Most often, a symbol supports a theme in the story. The theme I believe is supported in this short story is violence and cruelty. 

One source notes...

"Martin", Bobby’s surname, derives from a Middle English word signifying ape or monkey.

The reference made in this case directs the reader's attention to the hidden animal (the uncivilized "ape") that resides within each of us. The "inner" animal is prone to "violence and cruelty;" also present then is the sense that civilized behavior and control of the more basic, inner self is what stops people from acting on the impulse to be violent or cruel.

In addition, Martin comes...

...[f]rom the Roman name Martinus, which was derived from Martis, the genitive case of the name of the Roman god MARS.

Mars is the name of the Roman god of war. 

In reading the beginning of Jackson's story, we learn about the "children" first:

The children assembled first, of course. School was recently over for the summer, and the feeling of liberty sat uneasily on most of them; they tended to gather together quietly for a while before they broke into boisterous play, and their talk was still of the classroom and teacher; of books and reprimands.

This segment suggests that the "children," who are closer to succumbing to their baser instincts because they are young and perhaps not as strong in repressing their less civilized impulses, are used to being controlled by adults. The feeling of freedom ("liberty") feels awkward; their talk is of school and reprimands (ways in which the youngsters are further controlled) in school.

The same restrictions are not, however, placed upon the students now that school is over for the summer. They are not as aggressively restricted. It may then come as no surprise that...

Bobby Martin had already stuffed his pockets full of stones...

This could demonstrate Bobby Martin's tendency to "war-like" behavior.

The martin is a kind of swallow. It is said that the swallow was a...

...bird that cried “consolation” at Lord’s crucifixion.

It might, then, be said that the martin (swallow) is a bird associated with death.

Additionally, the swallow (in general) represents...

...One who is prompt and ready in doing business...

Bobby Martin is "ominously" prompt, pockets stuffed before any of the other children, in preparation for the event to come.

Symbolically (especially in literature), a reference to a thing or a color (especially something that is repeated, or one that is among a group of other symbols, e.g., the last names of characters in the story) takes on a deeper meaning for the reader when the symbolic significance can be tied to the action of the plot—bringing about a deeper meaning to the reader. As long as it can be realistically supported by information from the text, it should be considered a viable reference, one simply waiting for the reader to find a specific significance of the symbol to the author's overall message.

Additional Source:

http://fcweb.sd36.bc.ca/~sommerfeld_h/FOV1-001B8839/FOV1-001CF952/The%20Lottery%20Symbolism.pdf