Peter, your dictionary provides the year in which the term began being used, but no etymology.
The OED agrees with 1854 as the year when the term was applied to the card game. (The game itself is older; a version is described in Decremps in 1786. And, of course, the premise is essentially that of the Three-Shell Game, which goes back farther -- how much earlier is a matter of some dispute which can be debated at another time...)
Ah, but the word "monte" itself? Well, according to The Random House College Dictionary 1988 revised edition (which is rather dependable in my experience), the term made its way from Latin (mons, mont-) to Old English (munt) and Old French (mont), hence eventually the modern English word "mount" -- the noun, as in "mountain."
The application of the term to a gambling game has to do with a precursor to what is now played with only a few cards; the "mount" of "monte" was a heap of cards.