Some of the Greeks:

Achilles -> The Suffering of the People, The Grief of the People coming probably from the greek word "ἄχος" (achos) which means "pain" or "suffering" or "sighing" or "grief" and "λαός" (laos) which means "people". (Apollodorous gave us a different origin, that his name came from the greek α which is a compartment that means "no" or "non" and the word χείλος aka "lip" so it means to him "No Lips" because according to a myth he transfers Achilles was fed by Chiron with the marrow of lions etc and his lips never touched the breast of a nursemaid) The original name of Achilles according to Apollodorous was also supposed to be Λυγίρων Ligyron aka "the whining one" or "the crying one"

Odysseus -> The Anger Bringer, The Hate Bearer, Hated, Furious According to the Odyssey coming from the verb ὀδύσσομαι which means "to be angry at" given to him by his grandfather Autolycus who was angry at many people and many people were angry at him

Menelaus -> The Anger of the People, The Fury of the Army Coming from the Greek μήνις aka "fury" or "anger" and the word λαός which stands for both "people" and "army"

Agamemnon -> Very Steadfast, Unbowed, The One Who Provides Much, The One Who is Much Thoughtful Agamemnon's name seems to have a dual potential. One comes from the words "ἄγαν" which stands for "very much" or "grandiosely" or at some free translation "great" and "μέμνων" could indicate someone who "insists" or someone who "remains". The other is the verb "μένδομαι" which means "to think on" or "to provide" or "to be mindful".

Ajax (Aias) -> The One of the Earth, The One who Cries 'Ah me!' Ajax or Aias as it is in the Original Greek has rather interesting roots. The two primary I found is the word αἶα which seems to be another word for γαῖα aka "earth" however from Sophocles "Ajax" we seem to get phonetic association of Ajax the Great's name from the exclamation αἰαῖ which translates as "ah me!" or "ah! ah!" and the verb αἰάζω which means "to cry"

Teucer (Teukros) -> The Trojan, The Builder, The Maker According to one interpretation the Greek name of Teucer comes from one of the tribes of Trojans named "τευκροί" so it seems that he simply was named to resemble the name of that tribe. Another is the word τεύκτωρ or τευκτήρ which means "the maker" essentially or "the one who builds"

Patroclus -> The Glory of his Father Coming from the Greek πάτερ aka "father" and κλέος aka "glory" (if someone brings a female counterpart that would be the famous name "Cleopatra")

Diomedes -> Divine Council, Divine Wisdom From the word δίος aka "divine" and μήδος "council"

Nestor -> The Traveler, The One Who Returns The name seems to potentially have roots to the verb νέομαι where the word νόστος (nostos) comes from. The verb means "to return" or in one way "to travel". Nestor seems to be one of the few kings that returned to their homeland after Troy without much loss along the way. He has also traveled more than once in his life if we take the myth that has him as one of the Argonauts

Antilochus -> Against Battle Cries, In Front Of Battle Cries The name of the son of Nestor comes from the compartment αντί which means "against" of "opposite" or "on the other side of" and λόχος which stands for "battle cry"

Machaon -> The Fighting One, The Knife-Holder The healer's name comes from either the verb μαχάομαι aka "to fight" or "to battle" or the word μάχαιρα aka "knife". Both attributes for his healing aspects (surgical knife and the fight for life) as well as his attribute as a soldier and fighter

Idomeneus -> The Fury of Ida Deriving from the name of mount Ida Ῑ̓́δη and μένος aka "fury"

Helen -> The Bright One, The Radiant One Helen's name deprives from the root Ελ which means "light" or "shine" similar to the first compartment of the name of Greece Ελλάς (Hellas)

Penelope -> Wild Duck, Wild Goose, The Weft-Like Woman Two possible origins for Penelope's name seem to appear. One is the word πηνέλοψ that stands for a predatory bird most likely a goose or a duck but one other seems to be the combination of the Greek πήνη "weft" aka the loop-like thread at the end of the weaving process and ὤψ which stands for "face" or "eye" or "look" aka "appearance"

Telemachus -> Away From The Battle, Distant Warrior The name deprives from the particle τηλέ which means "from afar" or "from a distance" and μαχάομαι which means "to fight".

Antinous -> Against All Reason, Unreasonable, Madman Depriving from the compartment αντί which means "against" or "opposing" or "opposite of" and the word νους which stands for "mind" or "reason"

Amphinomus -> Between Two Rules, Between Two Shares Depriving from the compartment αμφί which means "two things" or "two sides" and νόμος "rule" or νομή aka "share"

Eurymachus -> Of Wide Battle, The One of The Wide Strife From the adjective ευρύς "wide" or "spread out" and μαχάομαι "to battle"

Polites -> Citizen The Greek word πολίτης

Eurlylochus -> Of The Wide Battle Cry From the compartment ευρύς which means "wide" and λόχος which means "battle cry"

Peisistratus -> Persuasion Of The Army, Obedience Of The Army From the ancient Greek πεῖσις "persuasion" or "obedience" (coming from the verb πείθω "to persuade") and στρατός "army"

Euryclea -> With Much Glory From the adjective ευρύς "wide" "plenty of" and κλέος "glory"

Anticlea -> Without Glory From the compartment αντί which means "against" or "opposite" and κλέος "glory"

Laertes -> The Gatherer of People From the Greek λαός aka "people" and εἴρω which means "to gather together" or "to rise up someone"

Eumaeus -> The One Who Searches Well, The Good Seeker From the compartment εὐ "good" or "well" and probably the verb μαίομαι which means "to search" or "to seek"

Philoetius -> Beloved of Fate From the word φίλος "beloved" or "friend" and οἶτος "fate" or "destiny"

Ctimene -> The Acquired One From the verb κτάομαι "to acquire"

Neoptolemous -> New Warrior, Young Warrior From the word νέος "new" or "young" and the word πτόλεμος aka an archaic form of πόλεμος which means "war" (giving also the name to πτολίεθρον aka "citadel" or "city surrounded by walls" or "castle") (Name given at birth Πύρρος Pyrrhus aka "Fire-Like" or "Fiery Like Fire" or potentially "Red Like Fire")

Argos -> Swift, Fast, Quick From the homeric word that means such

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Some of the Trojans:

Priam -> The Bought One Priam's name is technically his slave's name according to Dares the Phrygian and comes from the verb "πρίαμαι" which means "to buy" when his sister Hesione gave her golden veil as a payment for her brother to Heracles (According to the same source Priam's birth name was Ποδάρκης Podarkes which means "swift-footed")

Hector -> The Holder, The One Who Holds Out, The One Who Possesses, The One Who Holds (Everyone Together) Hector seems to emanate from the verb ἕχειν which means "to hold" or "to have" or "to possess"

Andromache -> The Battle of Men Depriving from greek άνδρας aka "man" and μάχη "battle"

Astyanax -> Ruler of the City, The King Of The City Coming from Greek ἄστυ "city" and άναξ which stands for "king" or "prince" or generally "ruler"

Paris -> Youth, Beautiful Youth Most versions seem to agree on Luwian origin and the name Parizitis which stands for "youth". His nickname Alexander comes from Greek αλεξ which means "to repel" or "to protect" and άνδρας which means "man" so the name means "Protector of Men" and "Repeller of Men" which he got when he protected the shepherds that raised him There is some folk etymology on the Greek word for backpack "pera" so his name could also mean "The One Carried in a Backpack" in this case but most likely not entirely the case

Sarpedon -> Someone On High Power, The One who Sees High Power Possibly of Luwian origin (according to Dunford) sar-pēdan with sar standing for "up" or "on" unless you also add the possibility of a Greek addition to εἴδω which stands for "to see"