The English word “geyser” was adopted from “Geysir,” meaning “the gusher,” originally the proper name of a specific hot spring in Iceland. With time and general lack of understanding by English-speaking visitors, it became a general word for spouting hot springs.
The English word geyser was adopted from the Icelandic word Geysir, the name of one specific hot spring in the valley of Haukadal (or Haukadalur). One of the earliest English accounts of the word was in The Annual Register, a publication originally edited by Irish philosopher and political theorist Edmund Burke, in the 1760s. In that account, “Geyser” appears as the proper name of that specific hot spring, but with time and general lack of understanding by English-speaking visitors, it became a general word for spouting hot springs.
The name “Geysir” means “the gusher,” from the Old Norse geysa “to gush,” from the PIE root gheu- “to pour,” which is also part or all of the origin of other interesting words including alchemy (suggesting “that which is poured”), confuse (suggesting something mixed), and font (as in typography, and which I wrote a bit about here as it pertains to molten metal being poured into moulds for printing presses).
Etymonline says the general Icelandic words for spouting hot springs are hverr, “a cauldron,” and laug, “a hot bath,” though I do not speak Icelandic, so if anyone here does, I’d be interested to know how common each of those words is and the distinction between them.