Mnemosyne was a Titaness, the mother by Zeus of the Muses. Her name is the actual word for 'memory', and she was so named, as Liddell and Scott nicely put it, 'because before the invention of writing, memory was the Poet's chief gift'. By a rather devious and suspect route she became identified with the Roman goddess of money, Moneta, who actually developed out of a by-name of Juno. Perhaps the Romans found Moneta (meaning 'mint') a rather ordinary name, so wanted to give it a more exotic meaning.
The specific link between the names was made by the rather obscure third-century BC Roman poet and playwright (of Greek origin) Livius Andronicus, who translated and imitated Greek originals. In the beginning of his version of the Odyssey he explains that 'Moneta was the ancient Latin translation of Mnemosyne ... nam diva Monetas filia docuif ('for the divine daughter of Memory [Moneta] instructed').
Thus the 'divine daughter' of Uranus became the Roman mother of the Muses! Moneta's name has also been assigned a derivation from the Latin moneo, 'to remind', 'advise', 'solemnly point out', and Cicero tells a far-fetched story about a voice in an earthquake at Juno's temple 'advising' that a pregnant sow should be sacrificed.