For example, I'd like to have a color called "text-color" or one called "content-box-color". How can I save a color like this and then assign it to an element?
Check out the wiki on Inkscape swatches.
Also if this isn't what you're after, there is a free program called Colorpic which is a standalone program that can pick up a color from anywhere on your screen and save it in a labeled palette :)
I use it every day.
Swatches is definitely the way to go here and offers some very useful functionality. Once set up correctly, you can change one color setting and have the change instantly propagate to many different objects that use that same swatch color.
Here is a quick tutorial:
Another tip: to rename your swatches dive in to the XML editor and look in the element at the top.