What persuades the lynching party to give up their attempt on Tom's life in Chapter 15 of To Kill a Mockinbird?
In a word, "humanity". This is the chapter where Tom is in the prison and Atticus is sitting on the porch waiting for the mob. Jem, Dill, and Scout come even though they have strict instructions not to follow him. In this case, however, it is a good thing for Atticus and Tom that the kids did what they weren't supposed to do. Scout talks with one of the men--Mr. Walter Cunningham-- in the mob about his son...the two of them are school friends. The very conversation reminds the man that they are all living people and that Mr. Cunningham owes Atticus Finch for past favors. They are all connected and her speaking of Walter's son to him at this moment in time softens his heart. He orders the mob to leave the area, and no one is hurt. Scout doesn't realize how tense the situation could have been and how much danger she and her family was in, but Atticus and Tom do.
This scene is one of the most heart-wrenching and emotional ones in the entire novel: Scout shows up at the prison steps, and speaks with Mr. Cunningham about his son, Walter, with whom she attends school. In fact, early on in the novel, Walter joins the Finches for dinner and drowns his food in syrup. This leads to Scout's life lesson from Calpurnia, where she is told not to place her self above her company, even if they are poor.
This prior experience with Walter Cunningham becomes invaulable as Scout relates to Mr. Cunningham, and turns an entire angry lych mob into a group of soft-hearted daddies and husbands. The black-and-white movie version of the novel does this scene, in particular, great justice.
Scout talks to Mr. Cunningham about his son Walter and how she attends school with him. One time Walter joined the Finches and drowned his food in syrup. When Scout calls him out for using so much syrup, Cal calls her out and makes her understand that you should not act higher than anyone, especially if they are your company. From here, she makes the lynch mob understand the wrongness they are committing and turns them into humans.