I took it in. The Genius had no idea what was wrong. They had never seen this particular error message before. He decided I probably had a corrupted file somewhere, and had me restore the device as new, and then have iCloud bring over my stuff.
This worked, in that the error message went away, but I didn't get my stuff back. None of my apps came back. I reinstalled them, but didn't have any data. They were also spread out across 10 pages of apps. None of the folders were retained.
To sum it up, it was a mess.
So I ended up restoring via iCloud (NOT as new, but from the iCloud backup.) This brought everything back, but I still had the error message. Then I found this discussion:
And followed alijm23's advice. Worked like a charm.