In Chapter 15 of To Kill a Mockingbird, when Atticus says, "Do you really think so?", what persuasive force is he relying on to change minds?
I think the important word here is "think." Atticus is suggesting that the men think about what they are about to do before they go through with their murderous intentions. As Atticus tells Jem in the subsequent chapter,
"A mob's always made up of people... Mr. Cunningham was part of a mob last night, but he was still a man." (Chapter 16)
Atticus is appealing to them as a friend and as a man, and he hopes that they will stop and think individually about their actions, rather than as a group--"a gang of wild animals." Atticus's "dangerous question"--"Do you really think so?"--is one he has used before with his children, and it is always meant to question the wisdom of an action, such as when Scout is about to make a rash move while playing checkers, and
"Bam, bam, bam, and the checkerboard was swept clean of my men." (Chapter 15)
Or when Jem misunderstands something, and Atticus gives him a reference book in order to find the answer himself. Atticus hopes that Cunningham and his friends will stop and think about the seriousness of their plans before making a decision they cannot reverse.
This is a good question, and the above answer does a good job, but there is more.
The sentence "Do you really think?" is used twice in the chapter. The more significant use of this sentence is in a conversation between Atticus and Mr. Links Deas. In this conversation, Mr. Links Deas tells Atticus that he is worried about the old Sarum crowd. He knows that they are racist and at times take matters into their own hands. For example, the mob, led by Mr. Cunningham, is the old Sarum crowd.
He told Atticus that he might want a change of venue. Atticus responds by saying "Do you really think?" The implication is that even if the old Sarum crowd is racist, they are not much more different than the others in Maycomb. All are racist. In other words, would a change of venue really change things? According to Atticus, probably not.