They're hoping that one person is so trusting that they will believe him. My mom would have been that person. She just absolutely would not have wanted to believe that this man blew his son's head off, would see his sniffling and crying and be so touched with the Paw-Paw nickname (Southern roots). She was absolutely impossible to convince once she made up her mind. She prayed on things and if God told her to forgive someone, she did.
If a case was simple, cut and dried, my mom could see her way to finding someone liable (she did once, in a case where someone backed their car out without looking and ran over an elderly person), but basically she said God said "Judge not" and she worried that if she judged other people, that things would go badly for her on Judgement Day. She never did much of anything wrong (although she had a short list of what she thought God would mention to her).
That's the hope here. It's a Hail Mary. However, my mom was also subject to peer pressure, so if 11 jurors find what the prosecution is about to do with this witness believable, it's possible my mom would have caved. It's also possible that she would have absolutely listened to her inner sense of things and refused to convict.