Iceland was not the island's first name. It was at least its third.
According to Wikipedia, the Norse explorer Naddodd reached it in the ninth century and called it Snæland, "Snowland." The Swedish explorer Garðar Svavarsson sailed around it and named it Garðarshólmur, "Garðar's Isle," after himself. Neither name lasted.
The name that did is usually credited to Flóki Vilgerðarson, the first Norseman known to go there on purpose. His story appears in the Landnámabók, a medieval Icelandic manuscript. He set out in 868 and brought three ravens to find land, which earned him the nickname Hrafna-Flóki, "Raven-Flóki." The third raven flew northwest and did not come back, so he followed it.
Flóki's settlement went badly. After a generous summer, he was not ready for the winter. Climbing a mountain in the Westfjords, he looked down on a large fjord, Ísafjörður, full of drift ice. He named the whole land Ísland. Etymonline, dating the English name to about 1200, sums it up: "so called for its ice-choked fjords."
Not every historian accepts the Flóki story. In 2016 the historian Gunnar Karlsson argued that the name came from sailors, not one disappointed settler. He pointed to a medieval Norse text describing Vikings sailing between Norway and Ireland who used Iceland's glaciers as a navigation mark. On his reading, they named the land for those glaciers. "When they saw these glaciers to the west, then they knew they had sailed too far," The Reykjavík Grapevine quoted him as saying.
The meaning itself is not in doubt. Ís is "ice," and land is "land." What is uncertain is who first applied it and why.
The name is often paired with Greenland. The Saga of Erik the Red says Erik picked an attractive name for that land to encourage colonists, after he was banished from Iceland. One land got a bleak name, perhaps honestly. The other got a sales pitch.
Permanent settlement is dated to 874, when the Norwegian chieftain Ingólfr Arnarson built his homestead where Reykjavík now stands.