An idol statue is an object of cult for the natives, which can be brought back to England to be sold for Funds or donated to the museum for Fame. Also there is a quest called Obtain Idol, where a collector of sorts will reward the player if he gives the treasure to him, but is not necessary that the quest is active in order to appear the idol.
A stone idol can be acquired by stealing it from a native village while resting there at night.
If the idol is present, it will be mentioned in the initial description of the village. Resting in there will give an option to try and steal it. Taking the worship object from the natives, will offend them, greatly lowering the Standing of the party.