"What songs the Syrens sang, or what name Achilles assumed when he hid himself among women, although puzzling questions are not beyond all conjecture."
– Sir Thomas Browne, "Urn-Burial"
There are two separate things going on with this epigraph: the first is what it says and the second is what it is. So let's start with what's going on in the passage before we start getting fancy with the larger context.
First off, let's talk about the Sirens (Syrens is an old-fashioned spelling). These creatures make up a threesome of fine mythological ladies (half-bird, half woman) that appears in Book 6 of Homer's Odyssey. The Sirens sing so beautifully that every guy who sails by them gets seduced into following their song. Still, as soon as these guys attempt to navigate their ships over to these fine females, they run aground and die on the rocky coastline of the Sirens' home. Odysseus (a.k.a. Ulysses), the hero of the Odyssey, manages to sail past this deadly island by stuffing his crew's ears with cotton so they can't hear the Sirens' fatal songs. He then ties himself to the main mast so he can hear (because he wants to know what it's like) but has no power to steer the ship astray.
Next up is Achilles – perhaps best known for being played by Brad Pitt in Troy. He's one of the main Greeks who fought in the Trojan War. The story goes that Achilles's mom, Thetis, heard a prophecy that her son was going to be killed at Troy, so to protect him, she dressed him up as a girl and hid him among the daughters of King Lycomedes (source). The plan didn't work because Achilles reveals his identity by picking up a sword out of a stack of objects meant to tempt him. But props to Thetis for giving it a shot.
So, back to the epigraph. This quote poses two questions that are impossible to answer. Who knows what songs the Sirens sang? After all, none of the men who heard the Sirens sing (aside from Odysseus, who isn't telling) survived to describe it. And as for Achilles's name, we don't even know if there was an Achilles, let alone what he might have been called while he was hiding from his fate. But, the passage is saying, even if we can't know we can still have fun guessing (i.e., conjecturing).
And isn't that kind of what the main character, Dupin, is doing in "The Murders in the Rue Morgue"? He's looking at an impossible set of clues and then coming, improbably, to the right answer. This epigraph is telling us that even the most obscure, ridiculous stuff isn't necessarily immune to human logic.
Now, as to that context we were talking about earlier – this is a passage from the fifth chapter of Sir Thomas Browne's 1658 extended essay "Hydriotaphia. Urn-Burial; or, a Discourse of the Sepulchral Urns Lately Found in Norfolk." We don't want to make this into a bigger thing than it really is, because we think the epigraph was probably mostly chosen for its mention of "conjecture," rather than a comparison with Browne's strange "Urn-Burial."
Still, broadly, Browne's work is about antiquarianism (a.k.a. archaeology or the study of old stuff). He's interested in why we keep trying to preserve human remains in urns and graves when what really matters is what we do, not what our bodies are. Aside from the philosophical points Browne's making, it's interesting to note that this epigraph is taken from a long essay about dead bodies. What with the whole "Rue Morgue" theme, we think that it would be fair to say that Poe's got corpses on the brain. This particular reference fits in to Poe's whole Goth-horror thing he's got going on.