Throughout the story, Day is always seen wearing his simple pendant necklace—it's something that he's very attached to and that he brings with him wherever he goes. The reason becomes clear later on when he has a flashback to the moment that his father brought home the pendant and gave it to him:
In the southern swamplands between the two warfronts. It's a genuine coin from nineteen-ninety. See the name? United States. It was real. (2.9.44)
It turns out that the pendant is a hiding place for something else—an old quarter (yes, as in twenty-five cents) that bears good old George Washington's mug on it. But this is a huge deal to Day's parents because it means that the United States of America once existed, that it's not something that's just a pipe dream or a myth. In this way, the pendant stands for the truth—the fact that justice and freedom used to reign, but the government doesn't want people to know about anymore.
The pendant also reveals the truth when June sees Day reaching for it in the middle of the night. It's this piece of jewelry that gives his identity away—she can see right away that he really is the criminal Day. In the end though, the pendant becomes a truth that they both share and believe in—that there's a better world out there, and that the Republic is not the kind of government that they want to follow.