What prized heirloom of Thorin's family does Bilbo hide?

The Arkenstone is the prized heirloom of Thorin’s family that Bilbo hides.

Bilbo discovers the Arkenstone in Smaug’s treasure hoard beneath the Lonely Mountain after the dwarves regain access to their former kingdom. The jewel is described as Thorin’s most treasured family heirloom and is especially important because it symbolizes the royal heritage of the House of Durin.

Bilbo secretly keeps the stone rather than giving it immediately to Thorin. When Thorin refuses to negotiate with the besieging Men and Elves, Bilbo gives the Arkenstone to them so it can be used as leverage in seeking peace. Thorin regards this as betrayal, even though Bilbo hopes to prevent a battle.

The Arkenstone is not a Palantír, a magical seeing-stone from Tolkien’s wider mythology, and it is not Bilbo’s Mithril shirt. Its importance is political and emotional: Thorin’s attachment to it exposes the greed and possessiveness threatening to destroy the alliance around the Lonely Mountain.