Receive the Everstone through an event
Head to Ilima’s house in Hau’oli City and enter the left room on the second floor to battle against Ilima. Defeat him to obtain the item.
Exchange for BP
You can get the Everstone in the Royal Dome by exchanging it for 16BP.
Obtain it from a wild Pokemon
A specific wild Pokemon will sometimes have a held item. You can capture the Pokemon to take its item or use moves such as Thief and Trick to steal the item from the Pokemon.
Also, when your Pokemon has the ability Frisk, you can determine whether the wild Pokemon has a held item or not, so this ability is rather convenient when hunting for items.
Geodude, Graveler, Roggenrola, and Boldore are the most likely wild Pokemon to have the Everstone.