The right tools will make all the difference.
1) Stranded wire. You want tiny wire that will tin easily. I don't have a recommendation. Wish I did.
2) The right solder. I'd use 60/40 or 63/37 tin/lead rosin core.
3) A soldering iron with as small a tip as you can find. Mine is the sharpest point here, which I think would be up to this job.
4) Good lighting.
5) Magnifying glass. I use those cheapo helping hands. Obviously there are fancier options.

Strip and tin the wire ends. Hold the wire onto the target spot. With a bit of solder on your iron, touch the wire and hold until it melts and the transfers enough heat to melt the solder on the pad. Hopefully the solder from the tip and the wire will become one with the solder on the pad.

Once the wires are connected, think about strain relief. I sometimes twist the wires together and put a blob of hot glue to hold things steady.

You just need the right kind of wire. a 50W station with a broad tip will do, magnification isnt really needed.

Do it like this:

• Hold the broad tip at a 30 to 45 degree angle, so the contact point is small.
• Melt up the solder joint on the PCB and hold fresh solder wire to it, so theres more solder on it, and some flux too.
• Remove the insulation from AWG30 wire wrap wire, and tin it freshly too (this is important actually).
• Push the wire on the added solder and just tap it with the iron so the solder melts up and now is covering both the joint and the wire. Pull it a little to see if the joint isnt cold (happens with this kind of wire).

 

Since this is going in a small rocket which will see a lot of g force and vibration, I would not use wire wrap wire or any solid wire. I would go with stranded wire between 26 and 22 Gage. Support (strain relief) the wire as much as you can, especially the ends.