Oh wow, this is such a cool question because it really digs into the heart of world-building for mythical creatures. A lot of writers seem to default to the standard 'lure sailors with song' power, which is fine, but it's the unique twists that make a siren OC memorable. I think it comes down to mixing the mythological source with a very specific personal history or a flaw in the power itself.
For example, maybe a siren's power isn't her voice, but the resonance of silence she can create, drowning out all other sound and causing panic. Or perhaps her song only works on people who are already hiding a deep secret, making her a walking lie detector with a deadly side. I've seen some amazing stories where a siren's powers are tied to a non-aquatic element—like a siren born in a desert who 'sings' the sand into glass or manipulates mirages. The 'how' often involves a backstory event: a deal with a different sea deity, a curse that mutated their natural abilities, or hybridization with another supernatural lineage.
The most engaging ones I've read always have a cost. The power to command sea leviathans might require the siren to permanently live in the crushing depths, never seeing sunlight again. It's that trade-off, the unique limitation, that makes the power feel earned and integral to the character's story, not just a cool add-on.