I opened the attachment in Noterize (one of the ten apps on the list). But the "Open In" menu in Noterize is the same as the list in Mail, except that it has only nine apps (missing Noterize itself, of course). Since none of these is GoodReader, this is no solution. So, I deleted three of the other apps (which I no longer use) that are on the "Open In" list and restarted the iPad. Now, the "Open In" list has a different set of ten apps, but still none of them is GoodReader (nor DropBox).
I checked several functions in GoodReader (upload, download, annotate), and all are working fine.
"Don't Open Anything" is "Off."
I deleted three more rarely used apps that appear in the "Open In" list, checked the list in Mail again (no restarts) and found that the replacement apps on the list now include both GoodReader and DropBox! Seems that the limit is ten apps and I have no control over which ones appear. I hope I don't ever need to reinstall any of the apps I deleted to get to this point.
Thanks to all for the help. Perhaps Apple will give us control over the "Open In" list in a future upgrade of iOS.