When the Brotherhood Without Banners arrives at the Trident, there must have been lots of dead bodies around the place, after the battles that recently happened there and the Red Wedding. I don't see why they took a particular interest in this woman's body — how did they recognize her, and why did Beric decide to resurrect her?
Then she surprinsingly became the leader of the Brotherhood and therefore changes their project, from being neutral and protecting the smallfolk from raiders (of the south and the north), to seeking Lady Stoneheart's personal revenge (against the Lannisters and the Frey).
In this way the Brotherhood Without Banners became the Brotherhood with a Wolf Banner (or a Trout), figuratively speaking.
How did they come by Catelyn and recognize her after being dead for several days in the water?
Why did they decide to bring her back to life?
Why is she in charge of the Brotherhood? Does it change the Brotherhood's project?