Okay, just so you know from the start, "Adonais" is not even who anyone might consider mourning in the episode. It's actually Apollo of Greek myth. Whoever came up with the title might have been thinking of the poem Adonais by Percy Bysshe Shelley (husband of Mary "Frankenstein" Shelley) written about the death of Keats or a different Greek god, Adonis.
Well, who knows for sure?
The whole point of the episode, if you can fathom one, is the Roddenberry god box getting its most literal iteration ever by using an established cultural deity directly in the story rather than inventing from whole cloth some being with unlimited power (such as Q, although that description/association would greatly distress Picard, so I wouldn't mention it to him).
Anyway, so "Who Mourns for Adonais" is basically the Star Trek version of the Greek adventure. Think of Kirk as Hercules for this episode (and thusly swell William Shatner's ego to even greater proportions). The story goes, the crew needs one of those helpful Starfleet historian types who only happen to appear when needed to verify someone the viewer knows but is presumably something people from the future might not know about (although Khan remains an exception even though he...should have done his nasty eugenics stuff by now and been shot into space sleeping).
The main thing to remember about the episode is that it's another keen reminder that the second season somehow amplified the whole concept of the series. I guess it's no wonder that there was still so much trouble keeping NBC interested. If it thought Star Trek was hard to take seriously before, what about the fun house mirror version? (Speaking of which, Mirror Universe showing up in two episodes!)
This is not an episode you have to go out of your way to catch, but it's certainly one that helped form the reputation of the series. Chances are you know about it even if you don't think you do. It's one of those episodes.
Also, just the title has a peculiar legacy in the franchise, too. A Deep Space Nine episode used it as a callback: "Who Mourns for Morn?" (Everyone, that's who.)
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