I would like to expand on the panem et circenses quote. For this question I am drawn to this because "Hunger Games" and the quote are very similar as they both refer to using food and entertainment as means of control.
There is a dialogue in Mockingjay, chapter 16 which I think supports my view:
..., whereas in the Capitol, all they have known is Panem et Circenses"
"What's that?" I recognize Panem, of course, but the rest is nonsense"
"It's a saying from thousands of years ago, written in a language called Latin about a place called Rome," he explains. "Panem et circenses translates into 'Bread and Circuses.' The writer was saying that in return for full bellies and entertainment, his people had given up their political responsibilities and therefore their power.
I think about the Capitol. The excess of food. And the ultimate entertainment. The Hunger Games. "So that's what the districts are for. To provide the bread and circuses."
This explains the idea behind the Hunger Games.
However about the calling it "Hunger" I was not able to find a reference. My personal interpretation is that Hunger refers to inhabitants of the districts as opposed to well fed Capitols. Also similar to Gladiators which also were not citizen. I think there were quotes where Capitols looked down on people from districts as primitives who were obsessed/concerned with food - while in the Capitol they concentrated on "civilised" matters like arts, fashion and such.
Out of universe interpretations point out that Hunger Games may be a play on the original quote. It might actually be a strategy to keep the districts hungry and keep the disengaged only through the games. As the games also created rivalry between the districts this might also be reasonable.