Could the arc have been caused by bad wire? The outlet was only one
year old, but the wiring is very old. Someone suggested that the wire
is too old and needs to be replaced. If it is charred, can it just be
cut back and used again?

Without seeing I can't really answer that, but it seems unlikely.
Since the arc happened when you plugged the light in (right?) the problem
had to either be the outlet or the connection between the outlet and the
wire. A defective hot wire does not need anything plugged in to short, and
a defective neutral should not short unless it is completely broken.
Did the person who installed the outlet know what they were doing? If the
outlet is a year old and not abused, a bad connection sounds like the
culprit. Was the wire attached to a screw terminal or pushed through a hole
in the back. The holes in the back are very prone to failure, at least on
the $0.99 outlets.
If there is no visible damage to the wire, and it is not overloaded, it
should be safe; though some 70 year old wiring looks pretty dicey.

If you have enough wire to cut it back to a sound section, it ought to be
okay, though again, I can't see the wire so it might be prudent to get
someone to see it if you have any doubt.