You can buy tone arm wire - very light - only a few strands of copper.You can buy a soldering iron - at my age I might use a watchmaker's eye glass.
Tweezers would be useful.
In The UK Amazon sell a rewiring kit for £25 + £3 postage - somewhat expensive for a few feet of wire.
If you're a beginner at soldering then practice first - the common mistake all beginners make is too much solder and not enough heat.
Peccavi
April 2013