I had a rear flat at the end of the ride today. The repair was uneventful, but when I got back on the bike, the rear tire had a strange "dead" spot and I felt a bump every time the wheel spun to that spot.
Does anybody know what this is? I deflated and re-inflated the back tire and it only made it a little better.

Make sure your tube does not have a twist or kink in it.

also make sure the tire is completely seated. see if you can isolate that area and check it out

+1 on this. I'll bet the bead is not completely seated. Deflate tire a little and run thumb and fore finger along either side of the tire where it joins the rim. Don't ask me how I know this

High spots form when tires break cords too.