BYBLIS was the Naiad Nymph of a spring near the town of Byblis in Lykia (Lycia) or Karia (Caria) (regions of Anatolia). She was also described as the Hamadryad of an ancient ilex standing beside the spring.

Byblis was originally a girl who fell in love with her own brother Kaunos (Caunus). When she attempted to take her own life she was was saved by the Nymphs.

Parthenius, Love Romances 11 (trans. Gaselee) (Greek poet C1st B.C.) :
"From Aristokritos [mythographer C3rd BC] History of Miletos and the Foundation of Kaunos by Apollonios of Rhodes [epic poet C3rd B.C.]: There are various forms of the story about Kaunos (Caunus) and Byblis, the children of Miletos. Nikainetos [Alexandrian poet C3rd B.C.] says that Kaunos fell in love with his sister, and, being unable to rid himself of his passion, left his home and traveled far from his native land: he there founded a city to be inhabited by the scattered Ionian people. Nikainetos speaks of him thus in his epic:--
‘Further he [Miletos] fared and there the Oikousian town founded, and took to wife Tragasia, Kelaeneus' daughter, who twain children bare: first Kaunos, lover of right and law, and then fair Byblis, whom men likened to the tall junipers. Kaunos was smitten, all against his will, with love for Byblis; straightway he left his home, and fled beyond Dia: Kypros (Cyprus) did he shun, the land of snakes, and wooded Kapros too, and Karia's(Caria's) holy streams: and then, his goal once reached, the built a township, first of all the Ionians. But his sister far away, poor Byblis, to an owl divinely changed still sat without Miletos’ gates, and wailed for Kaunos to return, which might not be.’
However, most authors say that Byblis fell in love with Kaunos, and made proposals to him, begging him not to stand by and see the sight of her utter misery. He was horrified at what she said, and crossed over to the country then inhabited by the Leleges, where the spring Ekheneïs rises, and there founded the city called Kaunos after himself. She, as her passion did not abate, and also because she blamed herself for Kaunos' exile, tied the fillets of her head-dress to an oak, and so made a noose for her neck. The following are my own lines on the subject:--‘She, when she knew her brother's cruel heart, plained louder than the nightingales in the groves who weep for ever the Sithonian lad; then to a rough oak tied her snood, and made a strangling noose, and laid therein her neck: for her Milesian virgins rent their robes.’
Some also say that from her tears sprang a stream called after her name, Byblis."

Antoninus Liberalis, Metamorphoses 30 (trans. Celoria) (Greek mythographer C2nd A.D.) :
"Miletos [son of Apollon and Akakallis] built the city of Miletos and married Eidothee, daughter of Eurytos king of Karia (Caria). She became the mother of twins, Kaunos (Caunus) and Byblis after whom are named to this day the Karian cities of Kaunos and of Byblis.
Byblis attracted many local suitors, because of her fame, some from nearby cities as well. She did not pay them much attention since an unspeakable desire for Kaunos was driving her mad. Because she did all she could to hide this passion, she kept it from her parents. But daily she was being gripped by an even more unmanageable Daimon and one night she decided to throw herself from a rock.
She went to a nearby mountain and set about throwing herself off. But Nymphai, pitying her, held her back. Casting her into a deep sleep they changed her from a mortal to a deity, into a Nymphe called a Hamadryas [or Hydrias] named Byblis. They made her their companion and sharer of their way of life. The stream which flows from that rock is called to this day by local people the Tears of Byblis."

Pseudo-Hyginus, Fabulae 243 (trans. Grant) (Roman mythographer C2nd A.D.) :
"Byblis, daughter of Miletus, out of love for Caunus her brother, killed herself."

Ovid, Metamorphoses 9. 450 ff (trans. Melville) (Roman epic C1st B.C. to C1st A.D.) :
"[Miletos] built the battlements that keep their founder's name; where, as she strolled beside Maeander's winding banks, her father's stream, that turns so often back upon its course, he joined in love a Nympha of beauty rare, Cyanee, who one day bore him there Byblis and Caunus, twins, a tragic pair.
The tale of Byblis shows that girls should love as law allows, Byblis who lost her heart to great Apollineus [Kaunos, Caunus, grandson of Apollon], her twin brother. Hers was no sister's love; her love was wrong. At first she failed to understand at all what her heart felt and never thought it sin to kiss him often or to throw her arms around her brother's neck, and long mistook the lying semblance of a sister's love. Then gradually her passion warped. She dressed with care to meet her brother, keen, too keen, to look her loveliest, and envious if ever some girl there was lovelier. She still had no clear picture of herself, her heart still formed no prayer, though inwardly passion burned high. She addressed him as ‘My lord’ and, hating words of kinship, wanted him always to call her Byblis, never sister. Even so she dared not let her waking thoughts admit her wanton hopes. But when she sank relaxed in quiet rest, she often saw the object of her love, and fancied too she lay with him and blushed even in her sleep . . . Troubled thoughts perplexed her . . . ‘What will become of me? Away, perverted passion! Let me love my brother with a proper sister's love! Yet if his love had first been fired by me, maybe his madness would have found me willing. Well then, if I were willing had he wooed, I'll woo myself. Can I speak out? Can I confess? Love will compel me! Yes, I can. Or if shame locks my lips, then I’ll reveal by private letter love my lips conceal.’ . . .
Poor girl, she passed all bounds, kept offering herself to his rebuffs, and soon, no end in sight, her brother fled, fled from his country and the scene of shame to found a city in a foreign land. Then Byblis was beside herself with grief. She beat herself in frenzy and tore down the tunic from her breast. Now openly she raved with no attempt to hide her hope of lawless love, and in despair forsook the home she hated and her fatherland to trace her brother, find that fugitive . . . Byblis ran howling through the countryside, watched by the wives of Bubasis, then on through Caria and Lycia she roamed, among the warrior Leleges, and now Cragus was far behind her and the streams of Limyre and Canthos . . . The forest failed; on the hard ground she fell, exhausted by her quest, and lay face down, with tumbled hair, among the fallen leaves. Often the Nymphae Lelegeides (of the Leleges) tried to cradle her in their soft arms and often sought to salve the fever of her love, and comforted with soothing words her heart that heard no more. She lay in silence, clutching the small sedge, and watering the greensward with her tears. And these, men say, the Naides made a rill, for ever flowing--what could they give more? At once, as resin drips from damaged bark, or asphalt oozes from the earth’s dark womb, or, when the west wind breathes its balm, the sun unlocks the water that the frost has bound, so, wasting by her weeping all away, Byblis became a spring. Still in that dale it keeps its mistress' name, still mournfully trickles below the tall dark ilex tree."

Sources:

• Parthenius, Love Romances - Greek Mythography C1st BC
• Antoninus Liberalis, Metamorphoses - Greek Mythography C2nd AD
• Hyginus, Fabulae - Latin Mythography C2nd AD
• Ovid, Metamorphoses - Latin Epic C1st BC - C1st AD

Other references not currently quoted here: Conon Narrations 2; Stephanus Byzantium "Byblis"; Nonnos Dionysiaca 13.546