I would solder a new wire. Doesn't take much to heat the wire up and pull it out, it takes a lot to damage a circuit board. Trick is getting the excess solder out so you can put the new wire in. Rework station is the easiest way, but a solder sucker or solder wick gets the job done.
Depending on the hole size and wire size you could heat up the hole to melt the remaining solder and slide the wire in. Cleaning and tinning the soldering iron with flux core solder right before touching to the hole is key.