<PERSON>, the author of the novels, is known to be a huge fan of ancient Roman and Greek mythology. For example, from this interview:
Q: Thanks to a cruel futuristic government, 24 children are chosen by lottery to compete in the annual Hunger Games—a fight to the death that’s televised live. How did you come up with that idea?
A: It’s very much based on the myth of <PERSON> and the Minotaur, which I read when I was eight years old. I was a huge fan of Greek and Roman mythology. As punishment for displeasing Crete, Athens periodically had to send seven youths and seven maidens to Crete, where they were thrown into the labyrinth and devoured by the Minotaur, which is a monster that’s half man and half bull. Even when I was a little kid, the story took my breath away, because it was so cruel, and Crete was so ruthless.
She discusses this in more detail in an interview with Scholastic:
Q: You weave action, adventure, mythology, sci-fi, romance, and philosophy throughout The Hunger Games. What influenced the creation of The Hunger Games?
A: A significant influence would have to be the Greek myth of <PERSON> and the <PERSON>. The myth tells how in punishment for past deeds, Athens periodically had to send seven youths and seven maidens to Crete, where they were thrown in the Labyrinth and devoured by the monstrous Minotaur.
Even as a kid, I could appreciate how ruthless this was. <PERSON> was sending a very clear message: “Mess with us and we’ll do something worse than kill you. 72
We’ll kill your children.” And the thing is, it was allowed; the parents sat by powerless to stop it. <PERSON>, who was the son of the king, volunteered to go. I guess in her own way, <PERSON> is a futuristic <PERSON>.
In keeping with the classical roots, I send my tributes into an updated version of the Roman gladiator games, which entails a ruthless government forcing people to fight to the death as popular entertainment. The world of Panem, particularly the Capitol, is loaded with Roman references. Panem itself comes from the expression “Panem et Circenses” which translates into “Bread and Circuses.”
The audiences for both the Roman games and reality TV are almost characters in themselves. They can respond with great enthusiasm or play a role in your elimination.
As a final note, this interesting page at Shmoop discusses some of the allusions to Roman society, including:
* The name Panem (as discussed above)
* The arena and its similarity to the Colosseum
* Cinna - there were two Cinnas associated with <PERSON>. One was a tyrant involved in his assassination, the other a poet murdered following the assassination after being mistakenly identified as the other Cinna.
* The large number of Roman names in the Capitol.
All of these references have obviously come from interviews with the author. The books don't provide any deeper reason for this and the movies certainly don't.
I would add that from an out of universe point of view, I think you've already described the link: "one could see the term "Capitol" and the actual Hunger Games as a kind of gladiator fights in relation to that". I think that's exactly what is at play here. The use of Roman symbolism throughout the Capitol only further emphasizes the strength and might of the Capitol over the weak, disjointed Districts. 72