What exactly is the significance of the dying elephant to George Orwell's thesis in the essay,"Shooting an Elephant"?
Good question. The elephant piles multiple meanings upon its back, like a beast of burden. Since Orwell mentions early in the essay that the British Empire is dying, we can assume that the elephant represents the empire and/or all colonial powers: ill, crazed, out of place, harmful to those around it.
It also represents the burden of Orwell, and of the colonial powers. He, and they, have to do unpleasant thing to beings alien to their experience.
The stubborn way in which the elephant clings to life represents the stubbornness of local culture, and, most simply, of biological life. Life is happening, and somehow the colonial/technological mechanisms of the British empire have to make sense of it, in this case by killing it.