The 'black box' used in "The Lottery" is a symbol to represent a bigger idea or concept behind this event. What does the black box symbolize?

The black box represents tradition and the people's unwillingness to break or change that tradition. It is noted that some have suggested using a new box. The barbaric ritual would continue, but simply with a newer looking box. But the consensus is that they wouldn't dare change anything about their traditions. So, the notion of a new box is squashed before it can even be debated. Tradition is more important: 

Mr. Summers spoke frequently to...

The black box represents tradition and the people's unwillingness to break or change that tradition. It is noted that some have suggested using a new box. The barbaric ritual would continue, but simply with a newer looking box. But the consensus is that they wouldn't dare change anything about their traditions. So, the notion of a new box is squashed before it can even be debated. Tradition is more important: 

Mr. Summers spoke frequently to the villagers about making a new box, but no one liked to upset even as much tradition as was represented by the black box. There was a story that the present box had been made with some pieces of the box that had preceded it, the one that had been constructed when the first people settled down to make a village here. 

The black box has deteriorated over the years. This symbolizes the deterioration of the ritual. That ritual is a remnant of the past. It had been initiated as some type of sacred ritual to appease God and/or be a sacrifice to promote a good crop. But note that other towns talk of getting rid of the ritual. This suggests that other places are becoming more logical, ethical, and scientific. The ritual always was useless and barbaric, but within the context of a more modern society, the ritual seems even more absurd and out of touch. It's tattered appearance symbolizes how it is a vestige of a less enlightened society. 

The black box represents traditional ways of thinking and people's fear of changing. Note that the villagers don't even know what the ritual is. The only things that they do remember are the stones: 

Although the villagers had forgotten the ritual and lost the original black box, they still remembered to use stones. 

This shows a thoughtless acceptance of tradition, traditional thinking, past ideologies and belief systems. The black box represents tradition and this notion of mindlessly accepting tradition, no matter what the case is.