Old English grighund (West Saxon), greghund (Anglian) "greyhound," probably from grig- "bitch," a word of unknown etymology, + hund "dog" (see hound (n.)).
The first element in the name apparently has nothing to do with color, as most of the hounds are not gray, but the exact sense of it must have been early forgotten, as it has been long associated with the color in popular imagination. In some Middle English forms it appears to be conformed to Grew, an old word for "Greek" (from Old French Griu). The Old Norse form of the word is preserved in Hjalti's couplet that almost sparked war between pagans and Christians in early Iceland:
Vilkat goð geyja
grey þykkjumk Freyja
[translation "I will not blaspheme the gods,
but I think Freyja is a bitch"]