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!
Aristophanes, lines 180ff.
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.