This image relates to a mysterious item, discussed at length in Section 8 of the Doctrine and Covenants, called "the gift of Aaron". The heading to Section 8 provides the context for the discussion:

Revelation given through Joseph Smith the Prophet to Oliver Cowdery, at Harmony, Pennsylvania, April 1829…In the course of the translation of the Book of Mormon, Oliver, who continued to serve as scribe, writing at the Prophet's dictation, desired to be endowed with the gift of translation. The Lord responded to his supplication by granting this revelation.

The revelation states, in relevant part:

6. Now this is not all thy gift; for you have another gift, which is the gift of Aaron; behold, it has told you many things;

7. Behold, there is no other power, save the power of God, that can cause this gift of Aaron to be with you.

8. Therefore, doubt not, for it is the gift of God; and you shall hold it in your hands, and do marvelous works; and no power shall be able to take it away out of your hands, for it is the work of God.

9. And, therefore, whatsoever you shall ask me to tell you by that means, that will I grant unto you, and you shall have knowledge concerning it.

10. Remember that without faith you can do nothing; therefore ask in faith. Trifle not with these things; do not ask for that which you ought not.

11. Ask that you may know the mysteries of God, and that you may translate and receive knowledge from all those ancient records which have been hid up, that are sacred; and according to your faith shall it be done unto you.

(D&C 8:6-11, emphasis added)

What is "the gift of Aaron"? The text provides several clues. First, Oliver has a history of using it, since "it has told [him] many things". Second, it is "the gift of God". Third, it is to be held in Oliver's hands (and kept there, impervious to any power). Fourth, it allows Oliver to "do marvelous works". Fifth, it is "the work of God". Sixth, the Lord will speak through it to Oliver and tell him anything he asks while using it. Seventh, it works through faith. Finally, it enables Oliver to translate ancient sacred documents.

With only these clues, the gift of Aaron remains very hard to identify. The task becomes much easier, however, when we look at the original version of the revelation as contained in The Book of Commandments, a predecessor volume to The Doctrine and Covenants, used by the LDS Church from 1831 to 1835. Section 7 of the book contains the same wording as the current version, except that the term "gift of Aaron" is replaced with the following language (in bold):

Now this is not all, for you have another gift, which is the gift of working with the rod: behold it has told you things: behold there is no other power save God, that can cause this rod of nature, to work in your hands…

What is the gift of Aaron? It is a "rod of nature".

What is a "rod of nature"? The historical record indicates that it is a divining rod (or dousing rod), commonly used to receive divine revelation, which Oliver Cowdery used as a youth to hunt for buried treasure. See e.g., Dan Vogel, ed., Early Mormon Documents (Salt Lake City: Signature Books, 1998), 1: 603-05, 619-20.  Cowdery's use of a divining rod to search for buried treasure evokes similar images of Joseph Smith hunting for treasure with a stone in a top hat. Oliver also wished to use his divining rod, in the same way Joseph Smith used his stone and hat, to translate ancient scriptures. Section 8 of the Doctrine and Covenants indicates that the Lord, through Joseph Smith, granted Oliver's request.

Additional information about Oliver Cowdery's rod is found here.