First of all, Islay is pronounced as Eye-la. Now we have that sorted out we can continue. There are several ideas where the name Islay comes from. Some think it comes from "Island divided into two" with Loch Gruinart and the Gruinart Flats being the dividing line. There are also ideas that it means "the law island", but is is also suggested that the name derives from a Pictisch princess called Ile, who lived around 650-700 AD. It is of course also possible that the name has emerged without obvious reasons.
According to Domhnall MacEacharna, who wrote the book The Old Parish Church Kildalton and The Lands of the Lordship: The Romance of Islay's names, the earliest known reference to the island comes in Adamnan's biography, Vita Columbae, of the Irish Saint, Columba, written in about 720 AD. St Columba visited Islay on his way north, prior to setting up the famous monestry on the island of Iona, of the south-west tip of Mull. Adamnan wrote it as "Ilea", describing it as an inhabited island, "Ilea insula habitabat", and also as "green, grassy Islay", a phrase which is still used in the Gaelic, "Ile Ghorm an Fheoir".
In a text in 740, it is spelt "Ili", while by 1095 it had become Yle. From then on, it is commonly Ila, Yla and Ilay. The present spelling was not widely adopted until about 1800. It is as if more modern writers were unhappy with Yla or Ilay and added an 's' to make it look more like the word "island". Islay is the anglicised spelling; in Gaelic the island is still spelt Ile.
Peggy Earl, writer of the book Tales of Islay, which is still available at C&E Roy in Bowmore has a different theory. Her favourite theory, however, concerned a Danish Princess called Iula, or Yula, who left Denmark with an apron full of stones of different sizes. As she proceeded on her journey some of the stones fell out, one becoming Ireland, another Rathlin and a third Texa. The remainder of the stones fell out and became the string of islands from Ardbeg to Kildalton. She perished in the soft sands off that coast and was taken to Seonais Hill above Loch Cnoc and buried there. What was described in the Statistical Account of 1794 as the grave of "a daughter of one of the kings of Denmark" is marked by two small standing stones about 10 meters apart, though there is no good evidence to support this tradition. Islay is said to have got its name from this lady, or perhaps she may have taken her name from Islay.