What gem does Thorin prize as the heirloom of his house?

The heirloom Thorin prizes as the treasure of his house is the Arkenstone.

The Arkenstone, also called the Heart of the Mountain, is a large, brilliant jewel discovered within Erebor. It becomes the treasured symbol of the Dwarves’ Kingdom under the Mountain and of the royal line descended from Durin. For Thorin Oakenshield, reclaiming it is more than a hunt for wealth: it represents the restoration of his family’s kingship and the recovery of his people’s homeland.

Bilbo finds the Arkenstone while exploring Smaug’s hoard and secretly keeps it. He later gives it to the besieging Elves and Men as a bargaining tool, hoping to prevent a battle over the treasure. Thorin’s intense attachment to the jewel also shows how possession and power contribute to his growing “dragon-sickness.”

The Arkenstone is not a Silmaril, Elessar, or Nauglamír. Those are distinct jewels or treasures from Tolkien’s wider legendarium. The Silmarils belong to the First Age, Elessar is a different green jewel, and the Nauglamír is a famous necklace.