Crying invokes a natural protection instinct in humans and animals alike; or, alternately, invokes the predator instinct of an easy lunch. Either way, mimicking the sound of a child crying is likely to draw the attention of either helpful or hungry marks. As long as there aren't many targets, it's unlikely to be detected, unless a whole gang approaches. Making "helpless" or "hurt" type noises is likely to bring food to your door, and has the added benefit of not being suspicious.
Certain sounds have been shown to have a direct effect on human emotions and physical state: Infrasound, very low frequency sounds, can cause a sense of awe or unease, and certain higher frequencies can cause nausea, irritation, vertigo, and disorientation. While these sounds aren't songs, per se, simply finding the resonant frequency of your target and blasting them with sound waves would likely incapacitate them enough to not be a problem. With a good set of lungs, it wouldn't be hard to take down a human for a few minutes with disorienting ultrasonic bursts. Of course, while loud sound can quickly disable a person, the sound will carry. If using loud noise, especially low frequency noise, make sure your sirens can escape with their target quickly, before anyone else shows up. Survivors are not likely to equate such a creature with the sirens from legend, however.
The right combination of chemicals secreted by the siren could cause any number of effects, from lowered inhibitions to hallucinations. It may be that the siren's song is merely to draw the gaze if those affected by a chemical substance, such as pheromones; when the affected target hears a "siren song", they turn towards the siren, draw in by motions and sounds she makes. Survivors will only remember a beautiful song.
The sirens of myth and legend drew sailors to themselves by their siren call, drawing them towards rocks that smashed their ships. Our siren may do just that, only at a higher technology level than mere music. Instead, the siren mimics the "ping" of SONAR, confusing sensors into believing there is danger (or safety) where there is none. With their sensors confused, ships wouldn't be able to spot a floating island, and would wreck; the siren could then further damage the ship, eventually sinking it. The more advanced the ship, however, the more difficult this would be; failing could draw a lot of unwanted attention.
Not all sound we hear is actual sound; interfering with the brain can cause it to register sounds without those sounds actually being present. The right electromagnetic waves could produce the sensation of music, as well as stimulating the pleasure center of the brain; those affected would claim to have heard a hauntingly beautiful song, but one that they cannot reproduce. While under the influence of the "siren song", they would be in a dreamlike state, unlikely to fight or even notice their surroundings. Survivors may even be tempted to return and find the source of the beautiful music; wearing headphones or noise-cancelling devices wouldn't stop the music, either, meaning detection is very difficult.