There are these little LEDs in this necklace, and attached to the LED is a long narrow plastic tube that is very luminous and transmits light easily
Those are called fiber optics, just like the link says in the title. Fiber optics can be made out of different materials and can be acquired and different places online and in stores, your probably looking for the cheap plastic fiber optics.
Im pretty certain the tiny fibers that you are talking about are indeed pieces of fiber optic cable. I have seen these before and am fairly certain of this. I am not sure where you can get it cheap.
