So we know that the Romans copied (almost) everything from the Greeks, who in turn copied (almost) everything from the Phoenicians, who in turn copied (almost) everything from the Egyptians and Mesopotamians, so where/when does the horsehair crest helmet first appear?
The earliest find we have of a hoplite-type crest is the helmet that was part of the Argos panoply, dating to c.725 BCE. Most of our evidence for helmets with crests comes from art, which always leaves a bit of uncertainty. Minoan and Mycenaean art shows helmets with various types of plumes and crests. One which might be similar to the later Classical horsehair crests is shown on a seal from Grave 518 at Mycenae dating to c.1500 BCE. The almost total lack of figurative art in the Aegean from c.1100-750 BCE means that it is difficult to tell if crested helmets were a continuous tradition in the region from the Mycenaean period down to Classical, or if they went out of use at the end of the Mycenaean period.
Most Mycenaean helmets do not seem to have been very similar to the Argos helmet, and so it might be that the horsehair crest of the hoplite was actually taken from Near Eastern models. At Carchemish, reliefs dating to c.900 BCE show soldiers with crested helmets. Finds of Urartian or Neo-Assyrian helmets of the same period also clearly show where crests were attached.
Good sources for this question are Snodgrass (1964) Early Greek Armour and Weapons, Everson (2004) Warfare in Ancient Greece, and Borchardt (1972) Homerische Helme.