Local mementos help you acquire items to use in your investigations. They can give you free Hints and match-3 boosters, as well as powerful boosters that double certain rewards while the booster is active.
You can find available mementos by tapping the bottom right-hand corner of the screen.
Tip: If you're missing an item to complete your memento, make sure to tap on the missing component and press "Find." This will take you to the map location where you have a chance of finding the necessary components.