The Economist's 1865 article on the assassination of President Lincoln refers to his death as the " most lamentable [event] which has occurred since the coup d'etat". Which coup d'etat would they have been referring to?

So it's obviously difficult to say with any certainty, given that the article doesn't give a whole lot of context to the specifics of the coup, but it seems likely that it refers to Napoleon III's coup against the Second Republic in France in 1851. Why, you ask?

Well, first, the context limits it to some time between 1815 (Waterloo) and 1865 (Lincoln's assassination). "since the coup d'etat, if not since Waterloo."

So just from there, you have a limited number of coups. The majority of them took place in Mexico, which experienced significant turbulence in the mid-19th century, with Santa Ana deposing leaders on a couple of occasions and being himself deposed. There was also the attempt to install a French puppet in Mexico in the early 1860s. But much as I love a good margarita, it seems unlikely that the author is speaking of one of the various Mexican coups. It seems even less likely that he is discussing one of the other more minor events, such as the Decembrist Revolution in Argentina.

Beyond that, however, a little more contextual evidence suggests that it refers to one of "the three most powerful countries in the world, America, France, and England." After all, Lincoln's assassination was lamentable because it affected all three of these nations, so it stands to reason that the aforementioned coup did as well. This leaves us with two real possibilities. The first would be the July Revolution of 1830, which saw the overthrow of the Bourbons and their replacements with the Orleanist Louis-Phillipe. The second would be the overthrow of the Second Republic, which was founded during the turmoil of 1848 by Napoleon III. To be perfectly clear, there is no good reason, beyond my own gut feeling here, that I'm arguing for Napoleon III's coup as being the specified coup, but it seems to me more likely that a publication such as the Economist, which has always had a very strong classical liberal point of view, would lament the overthrow of the Second Republic more than it would the overthrow of one monarchy for another. Furthermore, Napoleon III turned out to be somewhat of an adventurer during his reign. There was the aforementioned attempt to install a French puppet in Mexico. There was Napoleon III's somewhat duplicitous dealings with Piedmont during the Wars of Italian Unification, there was his work to basically tear down and rebuild Paris (or at least large parts of it), and I'm sure a lot more that I'm not as familiar with (not a historian of France, sorry). And this is all only the stuff that happens before Lincoln's assassination. He was intimately involved in German unification as well, although that came later.

So given the contextual clues in the article itself, I think that Napoleon III's coup of 1851 seems like the most likely bet to be the one mentioned in the article.

Sorry I don't have any sources to give you, but I'm not really sure how one would provide sources for this, since it's just a close analysis of a very small portion of the text.