The small statue appears to be a carved wooden idol, and is apparently important to the initiation ritual of the Children of Atom. Those who Atom deems fit to guide to the idol are accepted into the fold. The icon is most likely a representation of the shadowy figure known as the Mother of the Fog, who guides the Sole Survivor to the icon itself during the quest Visions in the Fog.
Separate versions of this object exist for each of the two associated quests. The player character can end up with both objects if they like. They have no weight and can be dropped from one's inventory to be used as decoration.