Sense 1 (“troop of soldiers picked to make an advance attack, or the first attack”) is a mistranslation of Dutch verloren hoop: verloren + hoop, mistaking the latter word for the homograph hoop ("hope"). Verloren is the past participle of verliezen (ultimately from Proto-Germanic *fraleusaną, from Proto-Indo-European *per- + *lewH-), while hoop is ultimately from Proto-Germanic *haupaz, from Proto-Indo-European *kouHp-nó- or *keHup-.

Sense 2 (“dangerous or hopeless venture”) is either an extension of the meaning of sense 1, or from the literal meaning of the words forlorn and hope.

• (RP) IPA: /fəˌlɔːn ˈhəʊp/, /fɔː-/
• (America) IPA: /fɔɹˌlɔɹn ˈhoʊp/

forlorn hope (plural forlorn hopes)

1. (military, archaic)
   1. A small troop of soldiers picked to make an advance attack, or the first attack; a storming party.
      • c. 1609–1622?, John Fletcher, “The Womans Prize: Or, The Tamer Tamed”, in Comedies and Tragedies […], London: […] Humphrey Robinson, […], and for Humphrey Moseley […], published 1647, →OCLC ↗, Act II, scene v, page 106 ↗, column 1:

        Ped[ro]. Stand to your guard ſir, all the devils extant / Are broke upon us, like a cloud of thunder; / There are more women, marching hitherward, / In reſcue of my Miſtris, then ere turn’d taile / At Sturbridge Faire; and I believe, as fiery. / Jaq[ues]. The forlorn-hope’s led by a tanners wife, / I know her by her hide; a deſperate woman: [...]

      • 1751, [Tobias] Smollett, “He is Concerned in a Dangerous Adventure with a Certain Gardener; […]”, in The Adventures of Peregrine Pickle […], volume (please specify |volume=I to IV), London: Harrison and Co., […], →OCLC ↗, page 49 ↗, column 1:

        [...] Pipes, who acted as the enemy's forlorn hope, advanced to the gate with great intrepidity, and clapping his foot to the door, which was none of the ſtouteſt, with the execution and diſpatch of a petard, ſplit it into a thouſand pieces.

      • 1901 May, Winston Churchill, “A Strange Meeting”, in The Crisis, New York, N.Y.: The Macmillan Company; London: Macmillan & Co., Ltd., →OCLC ↗, book III, page 433 ↗:

        He's the pride of our Vicksburg army. Not afraid of hell. A chivalrous man, a forlorn-hope man.

      • 1908 September – 1909 September, Jack London, chapter XIV, in Martin Eden, New York, N.Y.: The Macmillan Company, published September 1909, →OCLC ↗, page 117 ↗:

        He felt the stress and strain of life, its fevers and sweats and wild insurgences—surely this was the stuff to write about! He wanted to glorify the leaders of forlorn hopes, the mad lovers, the giants that fought under stress and strain, amid terror and tragedy, making life crackle with the strength of their endeavor.

   2. (obsolete) In the plural form forlorn hopes: the soldiers collectively making up such a troop; (by extension) a group of reckless adventurers.
2. (by extension) Any dangerous or hopeless venture.
   • 1794, Charlotte Smith, chapter XI, in The Banished Man. […], volume II, London: […] T[homas] Cadell, Jun. and W[illiam] Davies, (successors to Mr. [Thomas] Cadell) […], →OCLC ↗, page 226 ↗:

     From the tete-a-tete with Mr. Thomas Tough, ſhe goes to her deſk again, and begins to write "With what appetite ſhe may," in the forlorn hope of procuring from her bookſeller part of the money ſhe has been compelled to promiſe to the ſaid Thomas's peremptory demands, on behalf of Mr. Humphrey Hotgooſe.

   • 1842 December – 1844 July, Charles Dickens, “From which It will be Seen that Martin Became a Lion on His Own Account. Together with the Reason Why.”, in The Life and Adventures of Martin Chuzzlewit, London: Chapman and Hall, […], published 1844, →OCLC ↗, page 277 ↗:

     [A]s Mark, with all his vigilance, was unable to keep them [visitors] from the door; he resolved to go to bed—not that he felt at all sure of bed being any protection, but that he might not leave a forlorn hope untried.

   • 1876, George Eliot [pseudonym; Mary Ann Evans], chapter XL, in Daniel Deronda, volume III, Edinburgh, London: William Blackwood and Sons, →OCLC ↗, book V (Mordecai), pages 186–187 ↗:

     [I]t would be a cruel rebuff to a being who was appealing to him as a forlorn hope under the shadow of a coming doom.

• French: enfants perdus
• German: Verlorener Haufen
• Italian: fanti perduti, venturieri

• Italian: impresa disperata