I think that some terms aren't being used correctly:
- Stranded wire is wire made up of lots of really thin wires twisted together with no insulation between them. The bundle acts as a single wire.
- Solid core is just that: a solid metal wire.
- Twisted pair means two separate wires that have been loosely twisted together as a sort of shielding. They are insulated independently & can be solid or stranded.
The network cable is composed of four twisted pairs of solid core wire. You cut off the outer case & untwist the pairs to get a single wire. You can also leave the pairs twisted for a signal & ground pair.
Like davet said, plenum rated cable is the good stuff. It's also fairly pricey as it's hard to get in short lengths. I lucked into a nearly full spool a few years ago, so that's what I use.
Don't sweat it, though. You aren't going to be using much wire for your project & any hookup wire is fine for what you are doing.
Have fun & let us know how it works out!