This title seems to fit the story because the characters are exposed as "what they really are", rendered as a crystalization of their basic elements of personality and character just as materials are reduced in a crucible.
The play, in another way of speaking, presents a high pressure situation that reveals the truth of a person behind the social trappings, feints and postures of everyday life.
Great definition above, and it seems as if the entire story is a test of character. The Putnams' test is whether they will take advantage of the chaos and courtroom proceedings to further their own agendas regardless of the truth. They failed. The Court's test is whether it is able to administer justice. It failed. Reverend Parris's test is one of godliness, and he certainly fails. In fact, the entire episode is one of the great failures in American history. A few people in the story pass the test (Proctor, Elizabeth, Martha and Giles Corey, Reverend Hale, and Rebecca Nurse), but for most of them the series of events known as the Salem Witch Trials exposes the flaws and failings of their moral character.
I think that it is because Miller is trying (among other things) to look at the ways in which people respond to stressful times. As the previous post says, the word "crucible" implies heat and stress and change. Miller uses this title because he is interested in the way in which people's characters change when exposed to the sorts of "heat" and stress that we see in the play.
Let's start with the definition of the word. According to the Webster dictionary, a crucible is:
1: a vessel of a very refractory material (as porcelain) used for melting and calcining a substance that requires a high degree of heat
2: a severe test3: a place or situation in which concentrated forces interact to cause or influence change or development If you combine these three definitions, you come up with a container where under high heat (or in Miller's case...the stress of being accused of witchcraft (or communism under McCarthy)), there comes a severe test (the trials), where forces interact (the girl's false accusations and the people's fear about doing the right thing), to influence change (the hangings and murders of innocent citizens due to false accusations).
Miller chose this name because the play has a lot of difficulties, he want to show how some people acts with the problems, and chose the solution in tension, anxiety, and fear, even if they were wrong solutions.