The name Bombay duck denotes the bummalo (a small elongated fish of southern Asian coasts), especially when dried and eaten as an accompaniment to curries.

The first element of Bombay duck is an alteration, by association with Bombay (until 1995, the name for Mumbai, in India), of bummalo, which has also been spelt bummelo, bumaloe, bumbeloe, bumbalo. The following definition is from Hobson-Johnson: A glossary of colloquial Anglo-Indian words and phrases, and of kindred terms, etymological, historical, geographical and discursive, first published in 1886, by Henry Yule (1820-89) and Arthur Coke Burnell (1840-82):

(1903 edition)
Bummelo. A small fish, abounding on all the coasts of India and the Archipelago; Harpodon nehereus of Buch. Hamilton; the specific name being taken from the Bengali name nehare. The fish is a great delicacy when fresh caught and fried. When dried it becomes the famous Bombay duck […], which is now imported into England.

The first known mention of this fish is found in A new account of East-India and Persia, in eight letters being nine years travels begun 1672 and finished 1681 (London, 1698), by the English travel writer and doctor John Fryer (circa 1650-1733); when describing Bombay, he wrote the following:

On the backside of the Towns of Bombaim and Maijm, are Woods of Cocoes (under which inhabit the Banderines, those that prune and cultivate them), these Hortoes being the greatest Purchase and Estates on the Island, for some Miles together, till the Sea break in between them: Overagainst which, up the Bay a Mile, lies Massegoung, a great Fishing-Town, peculiarly notable for a Fish called Bumbelo, the Sustenance of the Poorer sort, who live on them and Batty, a course sort of Rice, and the Wine of the Cocoe, called Toddy.

The name Bombay duck is first recorded in Paddy Hew; A Poem, from the Brain of Timothy Tarpaulin (London, 1815), by A. Clark and William Combe:

To live there always on the rack
[…]
My carcase there did hourly waste,
Like Bombay duck and quite as fast,
When native strings him up by gills
And reeking fat runs down in rills,
Until it be reduc’d by sun
To shrivelled muscle, skin and bone.

It has been speculated that the second element of Bombay duck refers to the way the fish skim the surface of the water, like ducks. But, as the Reverend Abram Smythe Palmer (1844-1917) explained about Welsh rabbit in Folk Etymology: A dictionary of verbal corruptions or words perverted in form or meaning, by false derivation or mistaken analogy (London, 1882), it is more probable that Bombay duck

is one of a numerous class of slang expressions—the mock-heroic of the eating-house—in which some common dish or product for which any place or people has a special reputation is called by the name of some more dainty article of food which it is supposed humorously to supersede or equal. Thus a sheep’s head stewed with onions, a dish much affected by the German sugar-bakers in the East End of London, is called a German duck.

Similarly, in Britain, the red herring was variously known as Norfolk capon, Gourock ham and Dunbar wether (a wether is a castrated ram), and, in North America, Cape Cod turkey was a term for the codfish, Albany beef for the sturgeon, Taunton turkey and Digby chicken, or chick, for the herring.

Likewise, the following is from the above-quoted glossary, Hobson-Johnson:

The form Bombay duck has an analogy to Digby chicks which are sold in the London shops, also a kind of dried fish, pilchards we believe, and the name may have originated in imitation of this or some similar English name.