As I was looking into this to reproduce it, and I recorded my browser actions. In the process, I saw that when I started, the Custom HTML app had an "Auto Height" checkbox. I tried un-checking it, to see if it would make any difference, and when I edited the app settings later, the checkbox was completely gone.
I tried different browsers (Chrome and Firefox), a totally new page with unmodified custom app, clearing cache and cookies, everything I could think of. I tried starting Firefox in safe mode. If the gripper is shown, I can grab and drag it, but it doesn't "stick".
The latest thing I tried was disabling some Labs features. I disabled Windowed Dashboards, thinking it might have some interaction with the gripper. Magically, the gripper started working again! I re-enabled Windowed Dashboards, just to make sure, and sure enough, the gripper stopped working.
So for now, I'm turning off Windowed Dashboards and will tell my users to do the same. Please let me know if I should be expecting different behavior, or is this a bug in the experimental feature?
Thanks.