I make suncatchers using colored artistic wire, about 20 gauge. (Whatever the strongest wire is that will pass through my beads.) At one end of the wire make a hook by bending about 1" of the wire back onto itself, then fold that U in half so it looks like the hook of a hook & eye sewing notion. Then put on a bead, bend the extending wire at an angle or twist, curve, or whatever, then continue to add beads as you wish. At the end where you want to attach the crystal, thread the wire through the hole and make a flat spiral or curlique to keep the crystal in place. For ideas in using wire, go to the WigJig website. (If you can't find it let me know and I'll look up the address for you.)
If you did not want to use colored wire, and only want to "string" beads, I'd recommend a strong stranded beading wire (like Beadalon or tigertail) and secure the ends with crimps. You could form a loop at the end that would slip over the hook on a suction cup. Make a larger loop that would accomodate the "tail" being passed through it (like making a lark's head knot) if you wanted to slip this aroung the post of a rearview mirror, maybe. Or perhaps a small jewelry clasp that would connect back into the strand making a loop. Hmmm, maybe just make a 2" hook, bending it around a pencil to make the curve fit over the mirror post. The longer hook would balance better and probably not swing off.
I hope this makes sense and that I have given you some ideas. Let me know what works best for you.