Which names are important, and which are we supposed to be paying attention to?
Plutarch is obviously significant for who he represents in mythology…
Any others…
My personal theory are…
Snow and Coin - deliberately chosen for being the opposite of what they are “supposed” to be, but actually representative of who they are…
Prim - not sure of any deeper significance but “Prim and Proper” - or purity if you like
Peeta - phonetic (and alternate) is PITA - or Pain in the Arse, his moralising is a pain in the behind (and yes, I get the PITA bread reference)
Gale - literally a “tempest” - or uncontrolled emotion
Katniss (Katnip) - a drug that cats play with - a nod to the seemingly unexplained preoccupation of the Capitol with Katniss?
Plutarch was a real person.
Primrose and katniss are real plants.
The capital, Panem, is from the Latin for bread and specifically from its use in “Panem et circenses/bread and circuses”. Most of the power brokers in the capital have Roman or Romanized Greek names (e.g. Plutarch, Coriolanus Snow, Caesar Flickerman).
On the whole though I don’t she’s as playful as J.K. Rowling was as much as she liked the way names sounded. We don’t know a lot about Coriolanus Snow’s rise to power other than he used poison- his similarity to Shakespeare’s Coriolanus is that both were schemers, but where Shakespeare’s title character is consumed by personal animosity and vindictiveness Snow is actually the opposite. As he tells Katniss, while he’s not above killing children or innocent civilians he doesn’t kill for vindictiveness, just political gain.
Names of people in the provinces reflect what the specialty of that province is, most particularly Wiress and Beety.
You are overthinking the whole thing.