Athena

“Hear me, daughter of aegis-bearing Zeus, you who spy out all my ways
and who are with me in all my hardships; befriend me in this mine hour
(statement here)”
-Iliad

“This sort of a great-hearted overseer, a daughter of a strong-father
Holds her hands above our city, Pallas Athena (your request here)”
-Solon

“O Pallas, protector of the city, The most sacred city- and defender
of a land that surpasses all others in war and poetry (your request
here)”
-Aristophanes

“Queen, the foundation of the land and the city is yours, you are its
mother, mistress and guardian (your request here)”
-Euripides

Eileithyia

“Oh! goddess divine, Eileithyia, patroness of women in labour (request here)”
-Aristophanes, Lysistrata

“May merciful Selene hasten the tenth month for the bringing-forth [childbirth], but spare her, Eileithyia, I pray thee; and thou, O babe, spare thy mother, hurt not her tender womb and swelling breasts.”
-Statius, Silvae

“Eileithyia, come thou when (name here) calls, to bless her pains with easy birth.”
-Callimachus

Tyche

“And you Tyche, how many shapes you take, how you make playthings of the children of men! Be gracious, all-subduer!”
-Nonnus, Dionysiaca

“Daughter of Zeus Eleutherios, Tyche our saviour goddess, I pray (request here). For your hand steers the ships of ocean on their flying course, and rules on land the march of savage wars, and the assemblies of wise counsellors.”
-Pindar, Olympian odes

Nymphs

“O Anigrian nymphs, daughters of the river,
You divine beings who forever tread these depths
With your rosy feet, hail! And may you preserve
Cleonymus, who dedicated these beautiful images
Beneath the pines for you, o goddesses.”
-Anthologia Palatina

Primeval Eros AKA Phanes AKA Protogonus

“You also, Eros, primeval founder of fecund marriage, bend your bow, and the universe is no longer adrift. If all things come from you, friendly shepherd of life, draw one shot more and save all things.(request here)”
–Nonnus, Dionysiaca

Charites

“Come rosy-armed Graces, virgin daughters of Zeus (statement here)”
-Sappho

“Come now gentle Graces, and fair-haired Muses (statement here)”
-Sappho

“Charities of the golden-distaff, grant the fame that convinces mortals; for the god-inspired spokesman of the violet-eyed Muses is ready to sing the praises of Phlios and the luxuriant ground of Zeus Nemeios.”
-Bacchylides, Fragment 9

Edited version for general prayer:

“Charities of the golden-distaff, grant (request here)”

“Hear now my prayer, you Kharites three. For in your gift are all our mortal joys, and every sweet thing, be it wisdom, beauty, or glory, that makes rich the soul of man. Nor even can the immortal gods order at their behest the dance and festals, lacking the Kharites’ aid; who are the steward of all rites of heaven, whose thrones are set at Pytho beside Apollon of the golden bow, and who with everlasting honour worship the Father, lord of great Olympos. Euphrosyne, lover of song, and Aglaia revered, daughters of Zeus the all-highest, hearken, and with Thalia, darling of harmony (statement here)”
-Pindar, Olympian Ode 14

Zeus

“King Zeus, lord of Dodona, god of the Pelasgi, who dwellest afar, you
who hold wintry Dodona in your sway, where your prophets, the Selli,
dwell around you with their feet unwashed and their couches made upon
the ground (your request here)”
-Iliad

“Charioteer of the thundercloud with untiring feet, highest Zeus! Your Seasons, whirling to the embroidered notes of the lyre’s song, sent me as a witness of the most lofty games. When friends are successful, the noble immediately smile on the sweet announcement. Son of Cronus, you who hold Aetna, the wind-swept weight on terrible hundred-headed Typhon (request here)”
-Pindar, Olympic Odes

“Oh! Zeus, our Father! If you would deliver men from all the evils that oppress them”
-Pythagorean verses

“Zeus is air, Zeus is earth, Zeus is heaven, yea, Zeus is all things and whatsoever trancendeth them.”
-Aeschylus

“King Zeus, give unto us what is good, whether we pray or pray not;
But what is grievous, even if we pray for it, do thou avert”
-Plato

“Lead thou me on, O Zeus and Destiny,
To that goal long ago to me assigned.
I’ll follow and not falter; if my will
Prove weak and craven, still I’ll follow on.”
-The Handbook of Epictetus

“Now at my supplication, O Zeus, father of the Olympian gods, grant
that the fortunes of our house be firmly established, so that those
who rightly desire the rule of order may behold it. Every word of mine
has been uttered in justice. O Zeus, may you safeguard it!”
-Aeschylus, Libation Bearers

Apollo

“Lord Apollo, you know how to do no wrong; and, since you know this,
learn not to be neglectful also. For your power to do good is
assured.”
-AESCHYLUS, EUMENIDES

“Lord Apollon, you know how to do no wrong; and, since you know this,
learn not to be neglectful also. For your power to do good is
assured (your request here)”
-AESCHYLUS, EUMENIDES

“Hear me O god of the silver bow, you who protect Chryse and holy
Cilla, and rule Tenedos with your might (your request here)”
-Iliad

“God of the silver bow, thy ear incline,
Whose power incircles Cilla the divine;
Whose sacred eye thy Tenedos surveys,
And gilds fair Chrysa with distinguish’d rays!
If, fired to vengeance at thy priest’s request,
Thy direful darts inflict the raging pest:
Once more attend! avert the wasteful woe,
And smile propitious, and unbend thy bow”
-Iliad

“Hear me, Silverbow, Protector of Chryse, Lord of Holy Cilla, Master
of Tenedos, And Sminthian God of Plague! If ever I’ve built a temple
that pleased you. Or burnt fat thighbones of bulls and goats– Grant
me this prayer (your request here.”
-Iliad

Poseidon

“Hear me, Poseidon, who circle the earth, and do not begrudge me the
accomplishment of all these actions for which I pray you. (Your
request here)”
-Odyssey

“O Lord of Horses, you who hold sway over
Swift-traveling ships and over Euboea’s mighty
Overhanging crags, please grant to us,
Who pray to you (request here)”
-Anthologia Palatina 9.90

Ares

“Ares, holds as a fortress of the gods, the bright ornament that guards the altars of the gods of Hellas. I pray for the city, with favorable prophecy.”
-Aeschylus

Hermes

“May Maia’s son, as he rightfully should, lend his aid, for no one can
better sail a deed on a favoring course, when he would do so. But by
his mysterious utterance he brings darkness over men’s eyes by night,
and by day he is no more clear at all (your request here)”
-Aeschylus

Chthonic Hermes

“Supreme herald of the realm above and the realm below, O Hermes of
the nether world, come to my aid (your request here)”
-Elektra

Dionysus

“O Lord, with whom Eros the subduer And the dark-eyed Nymphs And rosy Aphrodite Play, you who haunt The high mountain peaks, I beseech you, come to me With kind disposition, hear And fulfill my prayer (request here)”
-Anacreon

Demeter

“Lady Demeter my queen I am your appliant. I fall before you as your
slave. (your request here)”
-curse tablet

“O demeter you nourish my soul make me worthy of your mysteries (your
request here)”
-Aristophanes

“Demeter, you who taught us to work the earth and provides for us so
bountifully (your request here)”
-Iliad

Artemis

“O Artemis, holder of Delos and lovely Ortygia, entrust your holy bow
to the laps of the Graces, bathe your pure skin in the Inopos (request
here)”
-Anthologia Palatina

“Huntress and archer, maiden daughter of zeus and leto, Artemis to
whom are given the recesses of the mountains,
(your request here) for so above thine altars will (your name) offer a
vapor of frankincense”
-Select Epigrams from the Greek Anthology

Nyx

“Hail, kindly Nyx, you who have given us great glory (your request here)”
-Aeschylus

“O Nyx, our dark Mother (your request here)”
-Aeschylus

Sun

“Oh Sun, that seest and givest ear to all things, Earth and Rivers
(your request here)”
-Iliad

Hecate

“Hecate who holds her torch on high daughter of night witht the deep
breasts (your request here)”
-Bacchylides

Fates

“I to high-throned Klotho (Clotho, Spinner) and her sister Moirai (Moirae, Fates) add this my plea (request here)”
-Pindar

“Aisa, Klotho (Clotho) and Lakhesis (Lachesis), fair-armed daughters of Nyx (Night), hear our prayers, you all-terrible deities of heaven and the lower world: send us rose-bloomed Eunomia (Good Order) and her bright-throned sisters Dike (Justice) and garland-wearing Eirana (Eirene, Peace), and (request here)”
-Stobaeus

“You mighty Moirai (Moirae, Fates), through the power of Zeus grant fulfilment in the way to which Dike (Justice) now turns (request here)”
-Aeschylus

Misc

“First, libations to Zeus and Hera for timely marriage
The second cup of mixed wine I serve to the Heroes
Third, a libation for blessing to Zeus, the Saviour.”
-Pindar