What is “normal strength”? I use the photographer addon which lets me set exposure as real values. Using sun lamp & sky texture (~400 and ~40 strength), checking against sunny 16 rule (anything outside in front sunlight should be correctly exposed).
If I now replace my sun&sky with an HDRI, I bring up the background strength until I get the “feeling” I want from it. If I know the HDRI shows an overcast, I’d change the exposure to 1/100s, ISO100, f/8. Note that many (most) HDRI have a clipped sun. If the shadow contrast is too little, you may want to supplement with an actual sun lamp, where you use a reflective sphere to watch where your sun is compared to the way toned down HDRI’s sun.
I’m not using a grey card at all. I check false color for blown out, and usually allow a bunch of stuff to be blown out (sky, lamps, strong reflections etc). If not, I’d adjust the light strength of what is causing the blowing out, keeping the main subject in the green (or under or overexposed if that is on purpose). I’m not a fan of “do nothing to the lighting when we can in CG but keep everything completely unclipped using local adaption” HDR tonemapping techniques.
But everything cgCody said holds true. Just wanted to express my way of thinking about “how to expose” from a photographer point of view.