Since The Hunger Games was released to the world over the weekend, I've seen and heard a few people – mostly folks who haven't read the books yet - ask why it's called the Hunger Games. Why Hunger and not just Games of Death? One person assumed the movie featured people starving, and it's a fair guess. As someone who's read the books and proclaimed their amazing-ness to the world, I was turned to for answers by a couple of those people. And I realized there's not a straightforward one. I can't recall any specific point in the books when anyone explained why the death match was specifically called The Hunger Games. As with many points in Suzanne Collins' series, it's left open to interpretation. In the context of the series, Hunger can have multiple meanings.
In The Arena
Getting food is key for tributes to survive in the arena. Without water and food, weapons and slaughter don't matter. Hunger equals weakness. The Careers hoard the food from the cornucopia as much for themselves as to keep it from the other tributes. People are easier to kill if they aren't strong. It's hunger that leads Foxface to the poisonous berries Peeta collected.
Starvation doesn't prove to be a very entertaining method of death for the crowds though, so the Gamemakers don't always permit it.
Everyone's Starving
The Capitol controls the food, and though some districts even grow food, they don't get to keep all of it. Most of the districts are effectively always hungry. It may not come across as well into the movie, but the folks of District 12 at least are starving. Some outright die of starvation. That's why Katniss goes hunting. In the arena, Rue talks to Katniss about a similar situation in District 11 (not in the movie) so you get the impression that's widespread.
Districts 1 and 2 might not be as hungry, but tributes from those districts usually win the games and are rewarded with food. Winning the games isn't just about surviving, it's about bringing a little affluence to your home district. If you win, everyone wins. The Capitol punishes all the other districts by letting them go hungry.
The Tesserae
Across Panem, children eligible for reaping can choose to take tesserae to help keep their families from starving. Since most families are struggling and in need of all the food they can get, many take this option if they can. The deal is you can have your name added additional times to the reaping, once per family member, in exchange for the same number of tesserae. The tesserae allows the family to collect oil and grain rations once a month for a year. The rations are meager, but they can make a difference between life and death.
This means people increase their odds for being picked as tribute in order to help feed their families. Though Katniss is only 16, her name is the reaping bowl 20 times. If she wouldn't have needed to get food for her family, she would have only had her name in five times at the most. Being hungry leads to a greater chance of being in the games.
The Bloodthirsty Capitol
The Capitol is hungry for violence, blood, and death. Like spectators of the gladiators, the citizens of Panem's Capitol practically salivate in anticipation for the games. They are entertained, they crave it. They see it as a necessary part of life. This may be the most simple reason but true nonetheless.
The Rebellion / Creation of the Games
The Hunger Games are a warning. It's an annual punishment for the uprising; the Games are a reminder that the districts are at the Capitol's mercy. The Capitol, President Snow, controls all. The government can force the districts to sacrifice two children every year and turn it into a sporting event. They hold power over everything, including all the supplies, and if people resist, they will go hungry. Hunger is a constant threat the Capitol holds over the citizens of Panem.
Though my personal opinion is that the name links back to the rebellion and that the Capitol controls the food, it's easy to make a case for most of these explanations. Maybe they all contribute to some degree. It's a tribute (ha, see what I did there) to Collins' complex world-building that all of these are on the table.