Here's what's generated:
__________________________________________________ ________
| 09:44:40 PM | Info | SuperDuper!, 2.8 (96), path: /Applications/SuperDuper!.app, Mac OS 10.11.4 build 15E65 (i386)
| 09:44:40 PM | Info | Started on Wed, Mar 30, 2016 at 9:44 PM
| 09:44:40 PM | Info | Source Volume: Nick's HD, mount: /, device: /dev/disk1, media: Nick's HD, interconnect: Internal SATA, file system: "Journaled HFS+", OS: 10.11.4 (15E65), capacity: 998.32 GB, used: 702.93 GB, directories: 274403, files: 2231363, ejectable: NO, ACLs: Enabled
| 09:44:40 PM | Info | Target Volume: LaCie quadra 1TB-SD!, mount: /Volumes/LaCie quadra 1TB-SD!, device: /dev/disk2s3, media: LaCie d2 quadra, interconnect: External USB, file system: "Journaled HFS+", OS: N/A, capacity: 1000.07 GB, used: 819.68 GB, directories: 274856, files: 2269049, ejectable: YES, ACLs: Enabled
| 09:44:40 PM | Info | Copy Mode : Smart Update
| 09:44:40 PM | Info | Copy Script : Backup - all files.dset
| 09:44:40 PM | Info | Transcript : BuildTranscript.plist
| 09:44:41 PM | Info | PHASE: 1. Prepare to Copy Files
| 09:44:41 PM | Info | ...ACTION: Preparing Nick's HD
| 09:44:41 PM | Info | ......COMMAND => Verifying the integrity of volinfo.database
| 09:44:41 PM | Info | volinfo.database OK
| 09:44:41 PM | Info | ......COMMAND => Enabling permissions on Dr. Marty's HD
| 09:44:42 PM | Info | File system user/group ownership enabled
| 09:44:42 PM | Info | ......COMMAND => Verifying that permissions are enabled for Nick's HD
| 09:44:42 PM | Info | Permissions on '/' are enabled.
| 09:44:42 PM | Info | ...ACTION: Preparing LaCie quadra 1TB-SD!
| 09:44:42 PM | Info | ......COMMAND => Enabling permissions on LaCie quadra 1TB-SD!
| 09:44:42 PM | Info | /dev/disk1 on / (hfs, local, journaled)
| 09:44:42 PM | Info | devfs on /dev (devfs, local, nobrowse)
| 09:44:42 PM | Info | /dev/disk4s2 on /Volumes/Seagate Backup Plus-1TB (hfs, local, journaled)
| 09:44:42 PM | Info | /dev/disk3s2 on /Volumes/LaCie quadra 1.5TB-TM (hfs, local, journaled)
| 09:44:42 PM | Info | /dev/disk3s3 on /Volumes/LaCie quadra 0.5TB-Media (hfs, local, journaled)
| 09:44:42 PM | Info | map -hosts on /net (autofs, nosuid, automounted, nobrowse)
| 09:44:42 PM | Info | map auto_home on /home (autofs, automounted, nobrowse)
| 09:44:42 PM | Info | /dev/disk2s3 on /Volumes/LaCie quadra 1TB-SD! (hfs, local, nodev, nosuid, journaled)
| 09:44:42 PM | Info | /dev/disk5s1 on /Volumes/Adobe Flash Player Installer (hfs, local, nodev, nosuid, read-only, noowners, quarantine, mounted by DrMarty)
| 09:44:42 PM | Error | GetMasterBlock: Error 16 from read; amount=-1, wanted=512
__________________________________________________ ________

There's a "Send to Shirt Pocket..." button, but clicking on it has no effect.

Well, Nick, I can't see how that can be the case, but I don't know what else I can do. It' seems clear your drive has failed, which is what the log is showing. I wanted the additional diagnostic information sent with the button to try to gather more evidence, but if you can't send it, what we have will have to do.

Your drive has failed. LaCie can likely replace it under warranty.

Very bizarre. The drive doesn't appear on my desktop, but Disk Utility "sees" it, and seems to do a repair when I run First Aid, returning an "Operation successful" window. (I took a screenshot—is there a way to post it?).

PS: The drive is very, very old (pre-iMac...way out of warranty).

Well... if it's very, very old, perhaps it's time to replace it?

Of course. It's just that its strange behavior is perplexing.

Do you have any recommendations for a replacement?

I still like the LaCie D2 line - I have one of their recent units here, and it's been a good drive.

This unit http://amzn.to/1Vj1KRU is a 3TB USB3 model, and while there are Thunderbolt models as well, they're far more expensive (and probably not worth it).

I like it too. (FWIW, I've always had good luck with the LaCie brand. I do recognize, however, that there is indeed a lot of "luck" involved, inasmuch as LaCie doesn't manufacture the platters, but buys them from Seagate, Western Digital, etc. What you get, therefore, pretty much depends on "timing": What's in LaCie's warehouse when it assembles its hard drives.

I have a 2TB Porsche Design (given to me by Apple when they replaced a defective iMac several years back) that has been serving me well. Any thoughts about the current 3TB version? (http://tinyurl.com/h25maw5)?

I've never used the Porsche drives. But it's just a case design. The d2 I like because the case (which dissipates the heat from the drive) is more massive and a better heatsink.