We don't have many facts from ADwD to guess on it though since Brienne is only mentioned once or twice and there was nothing that even winked at the right word. Oathkeeper, Winterfell, Catelyn, Sansa, Arya, sword, Ned, Shadow, Yes, Renly, Jaime,... So many options that have or haven't been explored but why would any of them make Stoneheart spare her? But a simple word, apart from Yes, has been bugging me for quite some time. Catelyn is mostly portrayed through the fact that she is a mother (letting Jaime go to save Sansa and Arya, staying with Bran day and night). I think that whatever Brienne said had to appeal to her emotions besides revenge and almost all of the words above remind Stoneheart of something to take revenge for, except Renly who ment nothing to her. So I was thinking Mother, or perhaps Children? We know that each time Thoros revived Beric, Beric lost some of his memories. It is safe to assume the same happened to Catelyn, but being a mother is something she didn't, couldn't forget. This is just a feeling I have though.
