I think she is having it made specifically for her.
It seems to be a clasp, sort of like a large lobster clasp or large spring ring clasp, and with a fairly large jump ring at the bottom. I believe that the hinge mechanism in the Paula Abdul attachment is like the hinge portion of a French clip ear wire, except heavier (so more durable) than an ear wire.
We can get something that works pretty much like this, but looks different. It's a trigger clasp with a medium/large attached jump ring at the bottom. You can order it with the jump ring pre-soldered, or open so you can easily put it onto a charm. (and if you're putting it on ์ - charms like hers, your next step is to learn to solder the rings shut after they're attached to the charm!)
A trigger clasp is basically a lobster clasp, but a bit different shape. It's shape should allow you to hook it into most sizes of charm bracelet links, or very easily attach to any type of cording.
--Polly