In Aristophanes' Frogs, Dionysus crosses a lake to reach Hades. The lake is where we first (and last) meet the eponymous frogs of the play, and Aristophanes does not name it. We do know, however, that it's the same lake Herakles crossed on his twelfth and final labour: bringing Cerberus back from the underworld.
Do we know the name of the lake? Do we know where it might be located? Does any of the Herakles stories give out any details about it?
It's probably Lake Lerna, a real (former) lake in the Argive plain. You might also know this lake from the Lernaean Hydra, the killing of which was one of Heracles' tasks.
The lake has disappeared now, but here was its approximate location. (Taken from Wikipedia.)
Smith's Dictionary of Greek and Roman Geography sums ups its relationship to this myth:
The grove of Lerna contained two temples, in one of which Demeter Prosymna and Dionysus were worshipped, and in the other Dionysus Saotes. In this grove a festival, called the Lernaea, was celebrated in honour of Demeter and Dionysus. Pausanias also mentions the fountain of Amphiaraus, and the Alcyonian pool (ἡ Ἀλκυονία λίμνη), through which the Argives say that Dionysus descended into Hades in order to recover Semele. The Alcyonian pool was said to be unfathomable, and the emperor Nero in vain attempted to reach its bottom with a sounding line of several fathoms in length. The circumference of the pool is estimated by Pausanias as only one-third of a stadium: its margin was covered with grass and rushes. Pausanias was told that, though the lake appeared so still and quiet, yet, if any one attempted to swim over it, he was dragged down to the bottom. Here Prosymnus is said to have pointed out to Dionysus the entrance in the lower world.
If you're wondering why Dionysus would have to cross a lake to get to the Underworld, the Greeks actually believed that many freshwater bodies of water descended down to Hades (many of which connect there).
You can read about some of the reasons for this in Julie Baleriaux's paper she gave at the Penn-Leiden Colloquium a couple years ago: Diving into the underground: Poseidon Hippios and the Arcadian Landscape
Others as early as the Suidas have put forth Acheron as river, which is possible, but I don't think it's likely considering Acheron is mentioned separately in the play (line 471). You'll run into people suggesting it, though.
The lake in The Frogs is the Acherusian lake (λιμνη Αχερουσιος), into which the river Acheron flows. The identity of the lake seems clear from the directions to the underworld given by Heracles in the play:
Dionysus No more of that, but tell me which of the roads will bring us quickest down to Hell, and one that’s not too hot, nor yet too cold.
Heracles How will you go?
Dionysus The road you went.
Heracles It’s a hell of a haul. Right off you’ll come to an enormous lake, a fathomless abyss!
Dionysus How will I get across?
Heracles In a little boat—just so big!—an aged mariner will take you over, and take two obols for your fare.
Aristophanes (405 BCE). The Frogs, lines 117ff. Translated by Matthew Dillon. Perseus Digital Library.
and from the name of the ferryman:
Xanthias What’s this?
Dionysus This? The lake, of course, the very one he mentioned, and now I see the boat.
Xanthias Me too, by Poseidon, and this one here is Charon.
Dionysus Ah Charon, Charon, cheery, cheery Charon!
So the lake is the body of water across which Charon ferried the dead to Hades, that is, the Acherusian lake. Aristophanes does not name the lake, but here are two sources that do:
Chorus O daughter of Pelias, farewell, and may you have joy even as you dwell in the sunless house of Hades! Let Hades, black-haired god, and the old man who sits at oar and tiller, ferryman of souls, be sure that it is by far the best of women that he has ferried in his skiff across the lake of Acheron.
Euripides (438 BCE). Alcestis, lines 439–444. Translated by David Koavcs (1994). Perseus Digital Library.
Now these streams are many and great and of all sorts, but among the many are four streams, the greatest and outermost of which is that called Oceanus, which flows round in a circle, and opposite this, flowing in the opposite direction, is Acheron, which flows through various desert places and, passing under the earth, comes to the Acherusian lake. To this lake the souls of most of the dead go and, after remaining there the appointed time, which is for some longer and for others shorter, are sent back to be born again into living beings.
Plato (4th century BCE). Phaedo 112e–113a. Translated by Harold North Fowler (1966). Perseus Digital Library.
The name “Acheron” was given to several rivers, and the name “Acherusia” to several lakes, that were imagined to connect to the underworld. Pausanias gives a couple of examples:
Among the sights of Thesprotia [in Epirus] are a sanctuary of Zeus at Dodona and an oak sacred to the god. Near Cichyrus is a lake called Acherusia, and a river called Acheron. There is also Cocytus, a most unlovely stream. I believe it was because Homer had seen these places that he made bold to describe in his poems the regions of Hades, and gave to the rivers there the names of those in Thesprotia.
Pausanias (2nd century). Description of Greece 1.17.5. Translated by W. H. S. Jones and H. A. Ormerod (1918). Perseus Digital Library.
Behind the temple of Chthonia [in Hermione in Argolis] are three places which the Hermionians call that of Clymenus, that of Pluto, and the Acherusian Lake. All are surrounded by fences of stones, while in the place of Clymenus there is also a chasm in the earth. Through this, according to the legend of the Hermionians, Heracles brought up the Hound of Hell.
