Heimaey is home to Eldfell (which translates to Mountain of Fire). The 660 feet high volcano could potentially be the cause of the Elephant Rock and many other formations like it around the island. The volcano is known to erupt without warning, forcing the inhabitants of the island to act fast with tactics that use sea water to quickly cool the moving lava before it can alter their home too drastically, destroy the town, or potentially hurt anyone.
A scenario such as this could have been the cause of the enormous rock that happened to be shaped just like an elephant. Mother Nature, you artsy lady, you. What a glorious sculpture.
You can visit Heimaey and the other Vestmannaeyjar islands by ferry, and all have hiking trails available including some leading to Eldfell. If this destination is officially on your travel bucket list, be sure to go during the summer for even more fun animal opportunities. Besides the amazing Elephant Rock, you might see Keiko, the whale from the Free Willy films as this was where he was actually set free, and also summer is when the island becomes populated by millions and millions of adorable puffins!