Mrs. Farquhar plays a significant role in the story even though she speaks no lines of dialogue. Interestingly, she serves to make Farquhar both a more sympathetic character and a less sympathetic character.
The fact that Farquhar wants to "fix his last thoughts upon his wife and children" makes him a sympathetic character. Readers understand that he loves his family and immediately feel sorrow for the widow and fatherless children that this hanging will leave in its wake. Similarly, in section II, when Farquhar is sitting "on a rustic bench" with his wife, he seems domestic and harmless, which again makes readers want to take his side. In the final section, when he sees his wife, "looking fresh and cool and sweet," stepping off the veranda to meet him, readers welcome that reunion with relief. His evident joy at seeing his lovely wife again tugs at readers' heartstrings.
However, upon further reflection, readers may come to like Farquhar less when they consider his wife. Readers may perceive just how much Farquhar risked when he conducted the operation against Owl Creek Bridge. He is "ardently devoted to the Southern cause," owns slaves, and longs for "the opportunity for distinction." These dubious motivations take precedence over the wife and children that he ostensibly holds in such high esteem. From that perspective, Farquhar was a foolish man indeed, and with this knowledge, readers may feel less horrified to know that he does not escape his fate after all.
The inclusion of Farquhar's wife in the story adds depth to Farquhar's character and may make the reader either more or less inclined to sympathize with his punishment.