I've followed the generous advice here and purchased the Xrite Color Checker Passport. I'm using it with PS CS6, not Lightroom, so I converted to DNG, created a profile, loaded and activated the profile. I see the helpful change in color with the custom profile, but what doesn't make sense to me is that when I click the white balance eye dropper on the middle gray patch the color changes. Wouldn't it stay the same ? How can I have two different looks that are accurate ?
Welcome to the rabbit hole that is colour management!
The profile that is created, by default, is a single illuminant profile, that is, it calculates all the colour values based on a pre determined D number/white balance/ºK, I can't put my hand on the value that X-Rite use for a single illuminant profile but lets say it is D65, or 6,500ºK. When you open your image, by default PS will use the As Shot WB number. If that As Shot ºK is different from the profiles 6,500ºK then your center white square will have different values. When you eye drop the center white square you will shift the profiled numbers in relation to that WB to profile difference.
The center white square will only be "correct" if the shot WB and the profile WB match. There is a way around this, you can profile two differently illuminated images and create a single, variable, profile, this will extrapolate the WB selected so your center square stays neutral across, in theory, any WB shot at. These profiles are called Dual-Illuminant profiles.
I find the single illuminant profiles work fine for most uses, the colour response is better (you lose that magenta caste) but you still have the artistic freedom of WB control for ambiance.
Hope this helps.