A while back, I heard that Facebook can predict when two people in a relationship are about to break up, based on statistical patterns in account activity prior to break-ups.

I'm not familiar with the Facebook APIs or what sort of account activity is statistically significant, but that would be a cool app that's very similar to this. Log in with Facebook, and show which of your in-a-relationship friends has the highest probability of breaking up. Obviously harder than a notification after the break-up, but prediction would be cool and useful.