I was experiencing the same issue; "This Photo Has Already Been Imported" LR 5. For over a year everything worked like a charm, then I started to see this issue. Oddly enough, it would only occur with certain photos from the same photo shoot. Long story short, in my case the culprit was that there were already files with the same name in the catalog. In troubleshooting, I changed the name of the file of one of the images with which I was having trouble; appended '007' When I tried import again, the file was no longer grayed out, and LR happily allow me to import.
What's odd, however, is that even if I unchecked, 'don't import suspected duplicates' LR would not allow me to import the duplicate files. Anyhow, I plan to change the setting in my camera to use sequential files names, and for good measure, I'll pick a different file naming convention that what I've used in the past. I'm an amateur, so the file naming convention isn't that important to me--I don't think.