To create my rings, I use two instances of the Ellipse tool. Then, I center the two circles, and use the Pathfinder tool to delete the intersection. I am left with a thick ring. I make some copies of the ring. When I rescale, the thickness changes. How can I preserve the thickness?
Don't use the Pathfinder. There's no way to preserve the "thickness" of the ring when scaled. The thickness of your ring is an intrinsic part of the shape you made so it scales exactly the same as the rest of your shape.. The thickness literally is your shape.
Instead use a single path with a stroke. You can then adjust your stroke as needed and you can check "Scale Strokes and Effects" in the Scale dialog:
Transform Palette > Scale Strokes & Effects
Also Uncheck align to Pixel Grid / And Don't have Re-align new objects to Pixel Grid checked.
But really, you need neither of these because all you should be doing is creating a circle with a stroke and no fill.
Also, should mention...
Keep the same stroke size for each different circle size. Then if you need to fill those strokes go to Object > Expand.