"...But why, DeeAnna? I started the experiment by completely dissolving the salt. ...."
Yes, but you dissolved it in plain water. When you added the NaOH to the salt-water solution, you changed the rules of the game. Just because a chemical is dissolved in one set of conditions doesn't mean it will remain dissolved if you change the conditions.
For example, one way to make a "rock" candy is to heat plain water and white sugar to form a super-saturated solution. My brothers and I used to hang a cotton string down into the container of sugar solution and wait. The temperature change and time will cause the sugar to recrystallize back into solid form around the string.
You're doing much the same thing with the salt solution -- you're making a near-saturated solution of salt in water, then adding the NaOH. In this case, it's the added lye, not the change in temperature, that is causing the recrystallization/precipitation, but the general idea is pretty much the same.