dunno the exact engineering description (we engineers call it music wire, too!). but it is hardened somewhat, typically black sometimes just steel colored and is made of steel of one type or another.....being steel it won't solder at all, so it's usefullness is limited.
Seriously, that's all I've ever called it: music wire. It's usually right there beside the K+S (?) brass wire.
emt49, I would drop by Atlas' website to find if the handrails are available separately. I'm pretty sure they would be, it's a common part to replace. good luck.