After I tried many different options (magnet wire, heat shrink on bare wire, ...) the solution is, in the end, rather simple: Stranded wire is ok if you glue it with cyanoacrylate. Just buy 10m of wire from ebay ($1.49), and use a black permanent marker to make the black wire you may want.

Ok, if you do lot of electronics, you can also buy two spools of 1000ft 30 AWG single core wire from ebay at $8.50 pieces. For this project you will use about 30ft, i.e. about $0.25 from your stock.