Make sure that the Zippo is clear of fuel. If it will light, it is not empty.
Leave the Zippo out without its casing in a well ventilated space to speed up fuel evaporation.
Remove the lighter from its casing, grabbing by the chimney.
Using either your fingers or the flathead screwdriver, remove the flint by unscrewing the flint spring at the bottom of the lighter.
Do this slowly, as the flint will have a tendency to shoot out from the lighter once it is all the way unscrewed.
Store the flint in the casing of the Zippo or some other place where it won't be lost.
Remove the felt pad from the bottom of the Zippo, being careful not to tear it.
The flint tube will be in the way, but you can rotate the felt around the tube while pulling out and the felt should come free.
Remove the cotton pieces from within the lighter and arrange them on your workstation as they were placed in the lighter.
If the cotton still feels wet with fuel, allow more time for the fuel to evaporate in a well ventilated area.
By keeping the cotton arranged in the same way it was packed into the lighter, you make sure you will be able to place the cotton pieces back in the same way later.
Remove the spent wick by pulling down with the needle nose pliers through the bottom of the lighter.
Utilizing the pliers again, feed the wick in through the chimney first.
Feed enough through so that the wick doesn't rise above the chimney.
Begin replacing the cotton, placing the last piece you pulled out in first and doing your best to replace them in the same position they were in before.
As you replace the cotton, weave the wick around each piece. This allows more wick to be inside the lighter, increasing its fuel capacity.
Replace the felt pad by gently pressing it back into place.
Replace the flint by putting it back down the flint tube and screwing the flint spring back into place.
Refuel your lighter and place it back into the case. Test it out! If all went well it should light right up. See iFixit's guide to refueling your Zippo if you need any help doing so.
Be sure not to spark the lighter while it is out of the case, since closing the case is typically how the flame is put out.