Yes, I would leave the original wire unless it really needed to be replaced. Isn't that part of the antique thing.
Antique usually means tubes. And tubes mean HEAT. And higher power resistors because they also make HEAT. Well, you get the idea. Much of the damage I have seen in older equipment has been due to heat. That includes the insulation on the wires. My first consideration would be just that. Teflon insulation would be the best choice. But it can cost as much as ten times as much. There are other types of high temperature insulation, just check the label or spec sheets.
Solid vs. stranded? Stranded is for places where you need it to bend and flex. Using stranded is to prevent the wire from breaking, not the components it is connected to. Solid is OK for inside a chassis where a run is usually the same year after year, only gathering dust. Copper is not a very good structural metal. It is not going to put much stress on anything. I wouldn't worry about that at all unless you stretch it like a guitar string.
Gauge? You probably need more than one. 20 or 22 gauge may be a good first choice but you will be getting others.
Oh, and bus wire. You will need some bare, tinned, solid wire in two or three gauges as well: 24, 20, and 16 gauge would be a good start.
One thing I have done to get a variety of stranded wire in different colors is to buy or salvage some lengths of multi-conductor cable and strip them for the wires inside. I was lucky to be able to pick up some lengths at my various places of employment when they were being discarded. They make multi-conductor in solid wire too but that is not as common. One source of solid conductor is telephone cable but it may have smaller wire gauges.