Rabbi Elchanan Samet has a book called Pirkei Elisha about all the Elisha stories in II Kings. In that book, Rabbi Samet is making one basic argument: there isn't a single Elisha story that is intended to show: "Elisha can do magic." Through a careful literary and historical reading of each story, Rabbi Samet tries to show how broad social implications and important religious issues emerge in each case.
Five chapters of Pirkei Elisha are devoted to this particular story. What follows is my interpretation of a few of the points that Rabbi Samet makes in his book.
Elisha, the Company of Prophets and Haechad:
This story of the ax is the last in a series of stories which describe the interaction between Elisha and “the company of prophets,” bnei neviim which are introduced in II Kings 2:
And when the sons of the prophets [bnei neviim] who were in view at Jericho saw him, they said, "The spirit of Elijah doth rest on Elisha." And they came to meet him, and bowed themselves to the ground before him. (v. 15 KJV)
The ax story found in chapter 6, like all the bnei neviim stories before it, are intended to capture the tensions and ideological differences between Elisha and this "company of prophets."
The story begins:
Now the company of prophets said to Elisha, "As you see, the place
where we live under your charge is too small for us. Let us go to the
Jordan, and let us collect logs there, one for each of us, and build a
place there for us to live."
He answered, "Do so."
Elisha's response tells us that Elisha is not happy with the plan. “Do so,” in Hebrew just one word: lechu, means, “go, and I'm not going with you.”
Then one of them said, "Please come with your servants." And he answered, "I will."
“One of them,” in Hebrew haechad, (literally, “the one”) begs Elisha and is able to convince him to join. In Hebrew, the words "live/dwell" and "go" appear seven times. This repetition reflect the tensions surrounding this move.
They arrive at the Jordan River and then:
“as one was felling a log, his ax head fell into the water;”
The ax-head fell into the waters of the Jordan and cannot be retrieved. Had this accident happened prior to their move, the company of prophets would have been able to pick up the ax head from the ground and continue with their work. This accident is a continuation of the tension surrounding whether or not Elisha and the prophets should be by the Jordan River in the first place.
The word used to denote the guy cutting the log is haeachad, the same word used to describe the guy who convinces Elisha to join them on the trip to the Jordan in the first place. The recurrence of haeachad implies the person who lost the ax head is the same person who convinced Elisha to come on the journey. The guy who lost his ax head immediately calls out to Elisha:
he cried out, "Alas, master! It was borrowed."
Haeachad lost a borrowed ax head and cannot afford to pay it back. Haeachad understands that this tragedy happened because Elisha doesn't approve of this plan in the first place. Haeachad pleads with Elisha saying: "it's not fair that I should be punished so harshly because you don't approve of this project."
Just like before, haeachad manages to appease Elisha and Elisha performs a miracle to return his ax.
Moving back to the Jordan:
Moving to the Jordan River is a big deal. In the previous story, the Syrian general Naaman was healed from leprosy by immersing himself in the Jordan River. Before Elijah is taken up to heaven in a whirlwind, him and Elisha travel from Gilgal to the Jordan River (II Kings 2:1-6). After Elijah goes up to heaven, Elisha splits the waters of the Jordan River (2:14) and then travels to Jerico until he finally ends up in Gilgal (4:38), effectively retracing Elijah's steps on his final journey.
Throughout these narratives, the Jordan River reflects a locus of spirituality which is disconnected from the everyday life of regular people. The tension that underscores the relationship between Elisha and the company of prophets stems from a question about the role of the prophet in relation to one's larger social context. Does the prophet belong with the commoners in Gilgal or in solitary-spiritual contemplation of God on the banks of the Jordan? This is a central ambiguity within these stories and Tanakh as a whole.