The common idea that Edison decided to electrocute Topsy the elephant (who had killed a trainer) in Jan 1903 to prove a point about electricity is wrong on several levels. One, it was officials in Luna Park who decided to electrocute the animal, and they required the help of the Brooklyn Illuminating Co. It was not part of the "War of the Currents," which AC had already won to the point that ; Edison was not in control of the Brooklyn Edison Electric Illuminating Co., which simply used the name because of his patents; and nobody consulted with him about this. (That is in some contrast to the experiments on whether electricity could euthanize small animals, which were in fact carried out at his order; Topsy, however, was not one of them, and it wasn't to prove a point as the Brooklyn EEIC was already running on AC.) Edison still owned the company that controlled film operations, of course, but they filmed anything of public interest, and this was expected to draw a crowd and was thus recorded. There is a TREMENDOUS amount of mythic history around Edison in print and on the Internet, which is made all the more difficult to push aside because he really was stubborn, hard-nosed, and litigious. But the Topsy episode was not something he had any hand in, and there is no record of him ever offering or being asked to provide an opinion on the proposal.

The Edison Papers Project goes into more detail on the matter. As a project that's been cataloguing and annotating papers from the various companies since the 1990s, they know what they're doing and have spent a good amount of time separating fact from fiction. Full disclosure: I know most of the people involved with the project and did some work with them, because the history of 19th-century technology is kind of a cozy field. [edit: As for when and how the link came to be accepted in popular wisdom, that's a stumper. I know the origins of the mythical Tesla-Edison feud and its errors, but not the Topsy story's transference.]