Morpheus is the Greek God of Dreams, the son of Hypnos. Erebus with Nyx (the Goddess of night/darkness), birthed Hypnos, (also the God of Sleep, as well as Sickness, and The Injured or Suffering - his symbol is the poppy plant, Hypnos is always depicted holding poppies or scattering their petals on the sick or injured). Thanatos is Hypnos' half-brother & the God of Death (some lore say they are twins. Thanatos is always pictured crying/mourning, waiting for Hypnos to heal the afflicted), who only takes humans his half-brother, Hypnos cannot save.

It's lore that on Battlefields (such as on the Battlefield of Troy), Hypnos and Thanatos were there healing and taking away souls who could not be helped - regardless of side.

Famous painting of Hypnos and his half-brother: "Sleep, and his half-brother, Death", 1874 (in this painting, Thanatos is on Hypnos' Island of Lemnos, where Hypnos brings humans who are suffering. On this island is "Lethe", the river of "forgetfulness"; if one were to take a sip from the river, their painful memories would vanish. In this painting, they're depicted in Hypnos grotto, on his bed made of ebony, and Hypnos is helping his half-brother, Thanatos (who does not take pleasure in the death of humans), as he would a human).

Another popular painting of Hypnos healing the sick

I love the story behind this whole family of Gods/Goddess, from Hypnos and Thanatos, to Nyx, Morpheus & Erebus. Erebus and Nyx have done terrible things according to lore, but after the birth of Hypnos and Thanatos, all of them helped humans. Especially humans who were suffering and could not find rest.

"Hypnos is the son of Nyx ("The Night") and Erebus ("The Darkness"). His brother is Thanatos ("Death"). Both siblings live in the underworld (Hades) or in Erebus, another valley of the Greek underworld. According to rumors, Hypnos lived in a big cave, which the river Lethe ("Forgetfulness") comes from and where night and day meet. His bed is made of ebony, on the entrance of the cave grow a number of poppies and other hypnotic plants. No light and no sound would ever enter his grotto. According to Homer, he lives on the island Lemnos, which later on has been claimed to be his very own dream-island. He is said to be a calm and gentle god, as he helps humans in need and, due to their sleep, owns half of their lives.[3][4]"

The stories behind this family of Gods are interesting; while other Gods play tricks on Humans, or could be persuaded by other Gods to their bidding for things (like power) in exchange, this family was resistant and tried to do what was best for humanity. However, Hypnos did get tricked by Hera into putting Zeus to sleep, so she could could trick Zeus while he was sleeping; according to the tale, she took advantage of Hypnos' good-hearted nature and feelings of loneliness and wanting for Eros (romantic love), so Hera promised him a daughter for him to marry and start a family with. Hera also tricked Aphrodite in this process, so she could charm Zeus to stop his war.

This is what gave birth to Morpheus.