Repairing disintegrated old 9V wire
You don't really need to worry about the wire gauge for Lego parts.
This source indicates that only the beefiest lego motors breach 1A, and even then, only while completely stalled. Most of those motors even have thermal protection systems built-in to prevent damage in the stalled case. So anything around gauge 20 or lower would do the trick.  

Make sure to get a stranded core instead of a solid core so it will be flexible.  

After you trim out the old damaged wire, slip on some cheap heat-shrink tubing over the remaining part before you twist/solder on the new wire. The heat-shrink tubing will reinforce and protect the new connections you made.