The Mysterious lamp is a reward for completing the Troll Stronghold quest. The player receives two lamps for completing the quest by speaking to Dunstan the blacksmith in Burthorpe, who calls them "family heirlooms...found by my great-great grandfather."
The lamp can be used to grant 10,000 experience in any skill over level 30.
This item replaced the Law talisman previously awarded for completing the quest. This update was because players were complaining that it was too hard of a quest for the law talisman. Jagex thought that this was a more appropriate reward for the difficulty of the quest. If the player has completed the quest prior to the update they can return to Dunstan for the additional reward.