Laudator Temporis Acti

Mary Eastman, Dahcotah: Life and Legends of the Sioux Around Fort Snelling (1849):

The scenery about Fort Snelling is rich in beauty. The falls of St. Anthony are familiar to travellers, and to readers of Indian sketches. Between the fort and these falls are the "Little Falls," forty feet in height, on a stream that empties into the Mississippi. The Indians called them Mine-hah-hah, or "laughing waters."

Henry Wadsworth Longfellow in The Song of Hiawatha (1855) repeats this folk etymology several times, e.g.

  Only once his pace he slackened,
Only once he paused or halted,
Paused to purchase heads of arrows
Of the ancient Arrow-maker,
In the land of the Dacotahs,
Where the Falls of Minnehaha
Flash and gleam among the oak-trees,
Laugh and leap into the valley.
  There the ancient Arrow-maker
Made his arrow-heads of sandstone,
Arrow-heads of chalcedony,
Arrow-heads of flint and jasper,
Smoothed and sharpened at the edges,
Hard and polished, keen and costly.
  With him dwelt his dark-eyed daughter,
Wayward as the Minnehaha,
With her moods of shade and sunshine,
Eyes that smiled and frowned alternate,
Feet as rapid as the river,
Tresses flowing like the water,
And as musical a laughter;
And he named her from the river,
From the water-fall he named her,
Minnehaha, Laughing Water.

Even today the Minneapolis Park and Recreation Board, on its web page for Minnehaha Park, claims that Minnehaha means "laughing waters."

The etymology is half correct: mni does mean water, but haha means falls or rapids, not laughing.

There are a few passages from ancient literature that refer to laughing waters. The first is Aeschylus, Prometheus Bound 88-91 (tr. Herbert Weir Smyth):

O thou bright sky of heaven, ye swift-winged breezes, ye river-waters, and multitudinous laughter of the waves of ocean, O universal mother Earth, and thou, all-seeing orb of the sun, to you I call!

ὦ δῖος αἰθὴρ καὶ ταχύπτεροι πνοαί,
ποταμῶν τε πηγαί, ποντίων τε κυμάτων
ἀνήριθμον γέλασμα, παμμῆτόρ τε γῆ,
καὶ τὸν πανόπτην κύκλον ἡλίου καλῶ.

Catullus refers to laughing waters twice, first at 31.12-14 (tr. F.W. Cornish):

Welcome, lovely Sirmio, and rejoice in your master, and rejoice ye too, waters of the Lydian lake, and laugh out aloud all the laughter you have in your home.

salve, o venusta Sirmio, atque ero gaude
gaudete vosque, o Lydiae lacus undae,
ridete quidquid est domi cachinnorum.

and also at 64.272-273 (tr. F.W. Cornish):

The waters slowly at first, driven by gentle breeze, step on and lightly sound with plash of laughter.

quae tarde primum clementi flamine pulsae
procedunt leviterque sonant plangore cachinni.