We'll be using an electrochemical process called electrolysis in a two chamber apparatus. In one chamber we will put a saturated saltwater solution and a positive electrode. In the other we will put pure water and the negative electrode. The two chambers are separated by a salt bridge composed of gelatin and table salt. While Jello brand gelatin can be used, we will be using Knox unflavored gelatin. The salt bridge allows sodium ions to pass between the two chambers.
NOTE: This is dangerous. Chlorine gas is extremely toxic and hydrogen gas is highly flammable. This should only be done outdoors in a well ventilated area. Avoid breathing the fumes which escape from the chamber.
When current is applied the sodium and chlorine ions are attracted to the negative and positive poles respectively. In the positive chamber the free chlorine combines into chlorine gas which escapes while the sodium ions travel across the salt bridge to the negative electrode. Meanshile in the negative chamber the pure water separates into hydrogen gas and an a hydroxide molecule. The hydrogen gas escapes and the sodium ions from the salt bridge combine with the hydroxide molecules to form Sodium Hydroxide in solution.
This process can be used to produce NaOH solutions of up to 40% or so concentration. This can be further reduced or dried to produce higher concentrations. I believe the "raw" syrup will be sufficient for soap making.
Traditionally the lye syrup made from ashes was boiled down until sufficiently concentration. This was tested using the egg test. If an egg was placed in the solution it should sink up to an area about the size of quarter (between 1/2 and 3/4 of the egg below the surface).
If you want to get all green about it use a solar oven to heat the water for making the saturation solution and use a small solar panel or solar battery charger for the electrolysis.