Contact cement does the trick!
I had the same problem with one of my strops and just fixed it this morning. I carefully lifted the flap and applied contact cement (rubber cement) to both the strop and the flap. Contact cement will remain elastic and still keep the leather flap firmly bonded to the leather. Now, let them air dry about 15 minutes and then use a drinking glass to smooth out the area. Working the glass back and forth across the leather. This takes all of the residual contact cement off the surface of your strop and firmly bonds the flap back down to the leather of the strop. After a couple of hours, I then sanded it lightly with some 400 grit sandpaper. The last step was to apply a thin coat of strop dressing and work it in with the palm of your hand. When I was finished my strop looked good as new and I honestly can't tell where the nick used to be anymore.