The noise of a siren came from the road beyond our games field.

“Why is it called a siren?” asked a student. “It makes a horrible frightening noise to warn people. Isn’t that the opposite of what the Sirens used to do?”

Yes—it is the exact opposite. The Sirens sang a beautiful enticing song to lure people. Homer does not explain; and the little information that he gives is only in Book 12 of the Odyssey. Circe warns Odysseus that the Sirens bewitch any mortal who approaches them. If any person comes too close and hears their music he will never return home to his wife and children. The Sirens sit in a meadow and all around them are the corpses of men.

More than that Homer does not say, and the other stories about these enigmatic beings—that they were of monstrous shape, that they drowned themselves after they failed to entice Odysseus, and so on—all come from the post-Homeric era and seem like attempts to embellish a myth that Homer preferred to leave in the realm of mystery.

ἀλλ᾽ ὅτε τόσσον ἀπῆμεν ὅσον τε γέγωνε βοήσας,
ῥίμφα διώκοντες, τὰς δ᾽ οὐ λάθεν ὠκύαλος νηῦς
ἐγγύθεν ὀρνυμένη, λιγυρὴν δ᾽ ἔντυνον ἀοιδήν:
‘δεῦρ᾽ ἄγ᾽ ἰών, πολύαιν᾽ Ὀδυσεῦ, μέγα κῦδος Ἀχαιῶν,
νῆα κατάστησον,  ἵνα νωιτέρην ὄπ’ ἀκούσῃς.
οὐ γάρ πώ τις τῇδε παρήλασε νηὶ μελαίνῃ,
πρίν γ᾽ ἡμέων μελίγηρυν ἀπὸ στομάτων ὄπ᾽ ἀκοῦσαι,
ἀλλ᾽ ὅ γε τερψάμενος νεῖται καὶ πλείονα εἰδώς.
ἴδμεν γάρ τοι πάνθ᾽ ὅσ᾽ ἐνὶ Τροίῃ εὐρείῃ
Ἀργεῖοι Τρῶές τε θεῶν ἰότητι μόγησαν,
ἴδμεν δ᾽, ὅσσα γένηται ἐπὶ χθονὶ πουλυβοτείρῃ.’
ὣς φάσαν ἱεῖσαι ὄπα κάλλιμον: αὐτὰρ ἐμὸν κῆρ
ἤθελ᾽ ἀκουέμεναι, λῦσαί τ᾽ ἐκέλευον ἑταίρους
ὀφρύσι νευστάζων: οἱ δὲ προπεσόντες ἔρεσσον.
Odyssey XII.181-191

But when with our swift advance we came within hailing distance of them, the Sirens saw the quick vessel near and raised their voices in high clear notes: “Come hither, renowned Odysseus, hither, you pride and glory of all Achaea! Pause with your ship; listen to our song.  Never has any man passed this way in his dark vessel and left unheard the honey sweet music from our lips; first he has taken his delight, then gone on his way a wiser man. We know of all the sorrows in the wide land of Troy that Argives and Trojans bore because the gods would needs have it so; we know of all things that come to pass on the fruitful earth.”
So they sang with their lovely voices, and my heart was eager to listen still. I twitched my brows to sign to the crew to let me go, but they leaned to their oars and rowed on…            Translation by Walter Shewring.

Meanwhile what of the modern instrument? The OED explains:

siren, n.
a. An acoustical instrument (invented by Cagniard de la Tour in 1819) for producing musical tones and used in numbering the vibrations in any note.
[It goes on to quote the Annual Register of 1820:]
‘The Syren, a new Acoustical Instrument… In consequence of this property of being sonorous in the water the instrument has been called the Syren.’
b. An instrument, made on a similar principle but of a larger size, used on steamships for giving fog-signals, warnings, etc. Also, more generally, a device which produces a piercing note (frequently of varying tone), used as an air-raid warning, or to signify the approach of a police car, etc.; the noise itself.