Myrddin, you might be confusing Midas' name with Minos, who is sometimes confusingly called the father of Minotaüros. This monster's real name was Asterios or Asterion, and he was nicknamed Minotaüros, "Minos' Bull." Pasiphai, the wife of King Minos of Crete, was the mother of Minotaüros, and she was the daughter of the sun-god Helios and the Oceanid Perseïs. Androgeneia, one of Minos' lovers, is also said to have borne him a son named Asterion, a distinct character from the Minotaur, who is, understandably, sometimes confounded with him because of the family ties (them being stepbrothers and all).
There are several less-known though interesting details about Midas, mostly involving his relatives. His father Gordios [Gordius] is the man who invented the Gordian knot, a legendary knot which could not be undone. A prophecy existed that whoever could untie this knot would become the ruler of Asia. Centuries later, Alexander the Great of Macedon solved this with his sword, cutting the rope into which the knot was bound, and since he did conquer Asia he apparently fulfilled the prophecy. The Phyrgian goddess Kybele [Cybele] is supposed to have been the mother of Midas, who is said to have inherited the throne from his father Gordios or to have emigrated with his people into Phrygia from Mygdonia, or from Lydia, or from Mt Pieria where he was a disciple of Orpheus.
The run-in that Midas had with Seilenos [Silenus] and Dionysos [Dionysus] occurred while the latter was on his famous war campaign against King Deriades of India. Seilenos gained his release from bondage at the court of Midas when he agreed to share a word of wisdom with the king, which happened to be the opinion that the best thing for mortals was not to be born at all, and the next best thing to die as soon as possible.
The daughter whom Midas unintentionally turned into a golden statue was called Zœ. The River Paktolos [Pactolus] in Lydia was a source of gold whose origin was believed to be the fact that this is where Dionysos told Midas to bathe in order to lose his power of turning everything he touched into that precious metal.
Midas also had two sons Ankhouros [Anchurus] and Lytierses. Once during Midas reign, a huge chasm opened up in the ground at Kelainai [Celaenae], a city of Karia [Caria], which was part of Midas' territory. An oracle told the king that if he threw his most precious possession into the abyss, it would close back up. Midas cast gold and silver into the hole and nothing happened. Seeing this, his son Ankhouros reasoned that a human life was the most precious possession of all, and so, taking leave of his wife Timothea, he rode his horse into the abyss, which immediately afterwards swallowed him and closed up.
Lytierses was Midas' illegitimate son but as his only surviving heir he succeeded him on the throne of Phrygia. He lived at Kelainai, where he was later killed by Herakles.
Philostratos says that Midas' donkey-ears were the natural result of some of his ancestors being satyrs.