One definition for "crucible" is a severe test or trial. This is the definition most often attributed to the play. There is a direction connection: within the play, the witch trials were very severe. Also, many characters underwent severe trials, specifically John Proctor, who not only was put on literal trial, but his integrity, morals, honesty, and convictions were tested.
Other definitions refer to a container where metals are melted down and fused; the high heats melt most metals, but not all. In this definition, we can see that many characters (Mary Warren, Goody Good and Osburne) "melted" under the severe heat and pressure of the witchcraft accusation. But not John Proctor. He was made of tougher metal. He signs a confession at first, but in the end "tears the paper and crumples it" and then says, "You have made your magic now, for now I do think I see some shred of goodness in John Proctor". Being under the severe heat and pressure of this trial, the "magic" he refers to, he is transformed into a stronger man who has withstood what so many others were not able to.
Because of the various definitions of the word, and its applicability not only to the events, but also the characters within the play itself, "crucible" is an apt word to choose for the play.
As defined in dictionary.com, one of the meanings of "crucible" is "a severe test, as of patience or belief; a trial.
A "crucible" is also "a vessel made of a refractory substance such as graphite or porcelain, used for melting and calcining materials at high temperatures."
Both these definitions add to our understanding of what is to happen in the play. We have a society undergoing a severe test: will they (and us --- remember, this is the McCarthy era, and Miller was intimately involved with the House Un-American Activities Committee) be influenced by an histerial witchhunt, or will reason prevail. There is also an individual in the play, John Proctor, who will be severely tested in "The Crucible" when he has to choose between his sense of integrity, his name, and his wife and their unborn child.
It is my suspicion that Miller saw himself in the crucible when he was pressed to reveal the names of his "fellow communists." It is helpful to remember that Miller wrote an adaptation of Ibsen's "An Enemy of the People" at about this time, and Dr. Stockman was faced with a similar choice.
Proctor, Stockman, Miller --- all facing the test of their lives.