So what would trigger memories in your characters? Favourite fictional characters of theirs? Tunes? What does your fictional world have in the way of entertainment that people would want to remember?
Equally does your fictional world discourage people remembering the past because there is a new government, maybe one that took over by force, that wants their citizens to focus on the future? What happens to those who persist in remembering and go public over it?
Or are memories implanted in people? If so, who does the implanting and how? Can the implanting process go wrong?
Of course favourite memories are different from person to person. Memories can cause conflict (if only between two people who remember things very differently. See the song I Remember It Well for a wry take on this). But what if those two people are of great importance in your fictional world? What could a fall out between them mean for that world?