Chosen Solution
The phone has had its screen replaced. It has also been water damaged. But! It has been repaired twice and has been working fine for weeks and now it suddenly powered off and is now stuck at the itunes restore screen. Tried restoring with multiple different cables on two different computers and it just keep coming up with error 4013. Help?!
Error 4013 is usually NAND related but can also be caused by a whole bunch of reasons. Considering your phone was previously water damaged, the error is probably a latent issue that just now appeared. Was the phone ever properly decontaminated (ultrasonic bath etc) or was it just dried…and heaven forbid, left in rice? Other than a good visual inspection (you may find some obvious issues) or changing the battery as Ben suggests, I think you should look for a pro that does water damage and micro-soldering repair.
Try another battery. It seems the 4013-4015 iTunes restore errors have been fixed a lot by replacing the battery with another one.
