UPDATE: I went to Apple Genius bar this evening. They also tried a DFU Restore and still got "Error (3)". They also claimed to have no idea what "Error (3)" was and that they knew less than we do as developers about that kind of stuff.
So, I put my 25 years of Semiconductor experience to work. I suspect that Error code (3) is likely a "write verify error" from the storage Flash that is generated during the install process. This would happen if a "bit" of the Flash "got stuck" as a "1" or a "0" either because of a "slow write" or a "slow erase". This is common in Flash technology. The "fix" is to "make the chips cold" as "cold" semiconductors will both "write" and "erase" easier than "warm" Flash semiconductors.
To verify, I put my iPad 3 in the freezer (should use a freezer bag to prevent moisture, but I didn't) for 30 minutes. After 30 minutes I removed from the freezer, plugged in my 30 pin cable immediately and started a regular Restore. IT WORKED! I then restored all my apps from the prior day's backup. A few did not restore sucessfully, so I had to manually delete and reinstall. All went fine and my iPad is running iOS 7.0.4.
I love physics! The problem could repeat itself in the future, but then again it may not. If this works for you, and I suspect it will, you can repeat the "freeze" process again. It won't work for years, but it will extend your iPhone/iPad life for a while.
Good luck all!