The Kassites (Babylonia, c. 1531 BC and until c. 1155 BC) named horses and were well-established in horse-breeding. Some of these names have been dated to the reigns of Burna-Buriash II (c.1359–1333 BC) and Kurigalzu II (c.1332–1308 BC). These names are among the few words we have of the Kassite language, and are given in An Early History of Horsemanship
Draught animals are indicated by their coats, sires by names. Mares
are not mentioned. Some lines may be mentioned as examples: "Z.2 One
pair, one bay and one black, sons of Samlu, son of Armi. Z.3 One
(pair), one chestnut son of Pikandi, and one black son of Halsi. Z.3
One (pair) chestnuts, sons of Nuhhuru".
As a side note, there is also a reference in Looking for the Aryans to the Kassites "who appear to have a horse-based tribal name".
Although not the oldest named horse, Kanthaka is worth mentioning as there is some information on him (though how reliable is debatable). Kanthaka, the favourite horse of Prince Siddhartha, later and better known as Gautama Buddha. Prince Siddhartha is believed to have been born no later than 480 BC (died 400 BC), but maybe as early as 563 BC (died 483 BC).
According to the article Kanthaka, Horse of Buddha,
Kanthaka was 18 cubits in length and accompanied prince Siddhartha in
all the major events. Donning a pristine white color it truly
complemented the royal stature of the prince. It was the most able
horse in the court of King Suddhodana.
"Panel showing Prince Siddharta parting from Kaṇṭhaka his horse and Chandaka his groom." Attrib: © Marie-Lan Nguyen / Wikimedia Commons
As to Kanthaka's death,
Leaving behind his wife, his baby son, and a life of luxury, Siddartha
escaped the palace riding on Kanthaka. When he got off his horse for
the last time to continue his life’s journey on foot, Kanthaka then
died of a broken heart.