I've been wondering about this for a while now, but my research hasn't turned up much. As an aspiring linguist, I find myself fascinated by the shift in meaning of words derived from Σείρηνες. The Sirens were always depicted in the classical era as either entirely human or half-bird in form. Yet somewhere between the age of classical Greece and what I would estimate to be the late Middle Ages, the Sirens became so thoroughly confused with what the Greeks themselves might've hypothetically called ιχθυονύμφαι that the modern word for "mermaid" in many European tongues (most notably, Romance) shows clear derivation from "Siren." We have Spanish and Italian "sirena," French "sirène," and Portuguese "sireia," to name some such cognates.
This is not the strangest example of semantic shift I've ever seen (for example, "silly" meant something like "blessed" in Old English), so I know this sort of thing happens regularly throughout linguistic history. Still, this particular case has my curiosity peaked. Was it the Sirens' mere proximity to the sea that caused the mythological merger, or was there something else at work here? When and how did the equivalence develop?
On a whim, I also looked up the modern Greek word for "mermaid," and it only added another twist to the tale (no pun intended). Lo and behold, at least according to the online dictionary at WordReference.Com, the word is "γοργόνα." This actually made me chuckle, 'cause the juxtaposition of Ariel and Medusa is highly amusing! Even if it turns out to be an example of convergent evolution, I can rest assured that I've chosen a very interesting field (linguistics) in which to pursue a Ph.D.!
On an equally interesting side note, I don't believe Hans Christian Andersen ever gave his heroine a name. I mention that only because I find mythological change almost as fascinating as linguistic change. It seems that some myths go through a succession of prestigious versions that far outshine concurrent alternatives, often to the point of causing retroactive amnesia in the popular mind. I suspect that, if you asked a random person in the street today what H.C.A. named the Little Mermaid, he/she would be very likely (depending somewhat on age) to say "Ariel" even though that was purely a Disney invention!