Let's start with Gower's description of Marina's upbringing in Tharsus. Turns out she's really, really good at weaving and sewing, and music:
to th'lute(Pericles 4.25-27). So the night-birds were singing away until the beauty of her own voice silenced them. My Arden Shakespeare glosses "still records with moan" as "always sings dolefully" – but surely there are other possible reasons for moaning at night-time, aren't there?
She sung, and made the night-bird mute
That still records with moan
Why, yes, there are. So let's turn now to the Canterbury Tales. The General Prologue opens with a description of the gentle, fecund period of spring. Chaucer passes from the quickening of flowers and crops to close observation of the animal kingdom, noting spring is a time when
smale foweles maken melodye,(9-11). That is, the effect of spring on little birds is they're singing and mating like mad all night. Of course in Chaucer this is all a build-up to the punchline, that spring is when humans likewise experience an irresistible urge: the urge to go on pilgrimages!
That slepen al the nyght with open ye
(So priketh hem Nature in hir corages)
So where Marina's grace allows her to silence the night-birds and their moan, we can see that working on two levels, the literal (she's musically gifted) and the bawdy (foreshadowing her effect on her would-be customers at the brothel). And her purity is such that all the stirrings of spring would doubtless only spur her to greater holiness.