That error usually indicates a problem with iTunes being able to connect to the update server. It's possible that the server is just overloaded with everyone trying to get the new update today, or it could be something on your system or network blocking access. Apple has some steps to follow here that may help:
If that doesn't help, try restoring from another computer, if you have access to one.
Hope this helps.
Checked firewall settings, nothing appears to be blocking itunes or any Apple diagnostics.
On iMac I see "preparing to restore" then "verifying restore" and--
"itunes recovery mode
itunes is restoring the software"
bar creeps along from left to right on imac
on phone display there is an apple and same bar creeps from left to right
bar reaches full distance on imac
the "verifying" bar creeps along from left to right on imac until full. This bar then goes stripey.
"Restoring iphone firmware" I next see on iMac
"The iPhone "iPhone" could not be restored. An unknown error occurred (-1)."
"iTunes has detected an iPhone in recovery mode. You must restore this iPhone before it can be used with iTunes"
I too had problems with the iOS 6.1 update for my iPhone 5. I'm running Windows 7 with iTunes 22.214.171.124.
The inital download of iOS 6.1 went through without a hitch. After the download, the software did an extraction, but failed right afterwards with the dreaded "The iPhone 'xxxxx' could not be updated. An unknown error occurred (3014)." Restarting my computer & rebooting the iPhone had no effect. The 3014 error would keep popping up right after iTunes had finished "extracting" (decompressing?) the iOS 6.1 image..
I read the problem was that iTunes couldn't contact gs.apple.com. So, I decided to turn off the Windows 7 firewall...and that worked. For good measure, I ended up also disabling my Antivirus just in case.
The iOS 6.1 upgrade went very smoothly after that, and completed (I think) in about 15 minutes or so.