Same thing (connect to itunes graphic) with my 4S and win7 PC laptop through a USB 3.0 port when it failed. Note I am fanatical about keeping everything up to date so it was nothing old affecting it. Here's what I did and it worked though was scary. Why? The Vs 5 to 6 update required me to restore also but it also wiped out all the apps on the iphone. Manually putting everything back in through itunes was was brutal. I considered that the Apple equivalent of a PC BSOD.
Before updating:
Plug laptop into power. You never know how long this will take.
Synch with itunes.
Make a manual backup of everything through itunes.
Do the update physically connected to the PC with itunes, not wifi:
It got as far as the connect to itunes graphic and failed. Leave everything connected.
I rebooted win7 and moved the 4s connector to a USB 2.0 port. Why? Read somewhere that timing issues with faster USB 3.0 can be a problem when updating iphone opsys. Forgot that until the failure reminded me.
Started iTunes. Be patient with USB 2.0. Watch TV or something.
Waited while the 4s connected and then it asked for a restore from the backup I just made prior to starting.
It restored and all was well exactly as it was prior to to the update.
6.1 was installed and all my apps and everything else was there. The backup restore was obviously using the 6.1 opsys which I verified.
Unlike the Vs 5 to 6 fiasco, this seemed to worked. Don't know if it will for you but hope it helps some.