One way to write an effective monologue based off of a character in The Crucible is to choose a complex character and explore the depth of their experience.
One complex character is Abigail. She plays such an important role in the drama's development. She is the reason why the trials reach such a fevered pitch. She initiated many of the accusations and stirred the emotionally contagious pot that Salem became. However, she simply disappears at drama's end. We really have no idea what has become of her. It might be very interesting to write about Abigail's life once she leaves Salem. A monologue that talks about what she experienced, whether she still thinks of Salem and John Proctor, or even what she hoped to gain out of leaving Salem could be very effective. It would delve into the thoughts and motivations of one of the drama's most important characters.
Another topic for an emotionally intense monologue would be Giles Corey. He occupied such an important role in Act III. His legal challenge to Danforth and Hathorne was significant to the drama. He is hauled off to jail and the only information we have about him comes from Elizabeth when she talks about his death. It might be unique to get into his mind while he is in jail. He knows that he will refuse to speak. Exploring the motivations behind his silence might be very persuasive. The monologue could even take place as he is sentenced. It could examine his thoughts as the stones are added upon him. There could be a delving into the physical pain he experiences as well as his emotional resolve to continue with it. His cry of "More weight" is synonymous with resistance in the face of intense pressure. Entering into this mindset might be very worthwhile in monologue form.
You can select any character from Miller's drama for an effective monologue. The idea would be to enter their mind and explore what they might have felt and experienced as a result of the emotionally draining witch trials.