My recommendation would be to do the following:
First: The wire is probably smaller than 20 AWG.... likely 24 AWG or even 28 AWG..... No matter really, but same size or larger (smaller AWG number) will work. Same size looks neater...
Use STRANDED wire not solid, as stranded is more durable and easier to route in this application.
Buy a small roll of the stranded wire and some 3/32" heatshrink. Cut the existing wires in a convenient place (at least an inch or two fom the connector, preferably in the center). Strip 1/8" insulation from both ends of cuts. Cut a length of stranded wire of the desired lenght and strip it at both ends as well (1/8"
. Cut the necessary number or 1/4" or longer heatshrink sections and thread them onto the new extension (two per wire). Now either twist the wire ends together or "merge" the strands. Merging the strands will provide a smoother, more aesthetically pleasing splice, but twisting is easier.
Using a 15-25 Watt soldering iron (tip properly tinned) and small diameter, rosin core solder, solder each of the connections. When soldering, place the iron's tip on the wire junction to heat it up. Then place the solder on the wire junction, NOT the iron... When the connection is at the proper temp, the solder will flow into the connection. When the connection is adequately "filled" with solder, remove the solder first, and then the iron, or the solder will stick to the connection. Heat it too much and the insulation will melt back from the connection.... Too little and you risk a cold solder joint. Practice soldering lengths of your stranded wire together first to get the hang of it.
After al connections are soldered, straighten out the connections if necessary/fold the solder tail parallel with the wire (if necesary). Then, making sure the connections are cool first, slide the heatshrink over each of the connections. Heat the shrink tubing until it shrinks tightly over the connections. A heat gun is preferable for this, though you can also use a hair dryer. Even a BIC lighter will work, if you are careful not to burn the tubing.
Verbose explanation, but it is easier than it sounds, and provides a much better appearance, and permanent connection. And if you are going to be playing around with computers, cooling, fans, etc, you are going to need a soldering iron, strippers, wire cutters, solder, etc eventually anyway....
Hope this helps,