The Gibbering Gemstone is a miscellaneous crafting material in Diablo III. It is used to construct the Staff of Herding, which is necessary to access Whimsyshire.
It has an Item Level of 1 and requires a character level of 1.
"It seems to be active but it's difficult to tell."
Acquisition
Map of possible cavern spawns
It can be found in the second level of Caverns of Frost, a dungeon located in the Fields of Slaughter (Act III - Machines of War, The Bridge of Korsikk is the nearest waypoint). Start sub quest Destroy Ballistae in Machines of War and you will be on the bridge, start moving north, Fields of Slaughter are there.
There is only one of two possible dungeons in the Fields of Slaughter at a time, one being the Icefall Caves, the other being Caverns of Frost. The caves can spawn in any of several possible locations on the map. It is strongly recommended to begin the chapter after as the ballistae for the quest also spawn on the cave locations, and it will eliminate 3 possible spots.
Ballista spawn points share the same locations as the Caverns of Frost/Icefall Caves. If Machines of War has been completed the ballista spawn points are still a possibility they will just be empty. At least one of the two possible dungeons will always be on the map regardless what quest is currently active. Waiting to regenerate the map until after completing Machines of War will not increase the chance the Icefall Caves dungeon will spawn. It will, however, mark three spawn locations with broken markers, making the search for the caves much more simple.
Also note that after the checkpoint after completing Machines of War, Tyrael will join you for a short time. If you exit and restart to recreate the Fields of Slaughter map Tyrael will not immediately rejoin your party. When you move on to the Rakkis Crossing bridge Tyrael will reappear in your party.
When you find the Caverns of Frost you will be looking for a unique Lacuni Huntress called Chiltara, that usually spawns in the second level of the cave. Chiltara does not always spawn, so you may need to clear the dungeon several times. When she spawns, she has a rare chance to drop the Gibbering Gemstone. The item will be account bound as soon as it drops.
It may be better to reload the map if Chiltara doesn’t spawn in Cavern of Frost level 2.
Trivia
The gem itself is a reference to the hysteria-inducing "Chat Gem" that was present in the Battle.Net lobby in Diablo II. The gem was initially blue, but would turn purple and state that the gem had been "activated" in the chat window above. If clicked enough times rapidly the prompt would instead say, "Moo." It had no other noticeable effect, leading to rampant fan speculation. A similar gem is located at the bottom of the Diablo III forums.
Bug
As of patch 2.0.1 Chiltara can't drop The Gibbering Gemstone <-- This was not a bug but instead the droprate had been lowered to +/- 0% - 5%, in patch 2.0.5 the spawnrate of Chiltara has been increased.