There are four major mentions of hope in this book, which is a lot in a world where children are fated to die. Each of our main characters gets at least one mention, and it says a lot about them.
The first time we see hope in conjunction with Lev is when he "feels his hope trampled beneath thirty pairs of scuffed shoes" (2.13.17). Um, that's grim. Not only does it foreshadow how Lev himself will be trampled later (he'll be fine), it shows us that Lev isn't the type to hope. Why should he? He plans on dying before his fourteenth birthday.
Risa, short for sonrisa, which means smile in Spanish, has a sunnier outlook on things. At the Graveyard, she "allow[s] herself the wonderful luxury of hope" (4.28.61), even though she later regrets it when the Graveyard is destroyed and Risa is taken away. "All her hopes of a future have been torn away from her again—and having those hopes, even briefly, makes this far more painful than not having had them at all" (5.48.18). Hope, then, is as much a source of sustenance as it is a source of pain.
It's not all hopeless, though. In the very last line of the book, we learn that Connor "At last […] allows himself the wonderful luxury of hope" (7.69.59. This ends this morbid book on a hopeful note, which is nice, but it also shows us that Connor—who like Lev, always expected to die and never had reason to hope—has changed. He has a reason to live. Risa is part of it, and even though we're not told explicitly, we bet she's rekindled her hope, too.