Restore/update problems with iPhone 5s

Try this, if you haven't... DFU Mode: Plug your device into your computer with a USB cable. Turn off the device. Hold the Power button for 3 seconds. Hold the Home and Power buttons for 10 seconds. Release the Power button but keep holding the Home button.

In addition to what's suggested above, i've run in to issues where itunes can't unzip the IPSW/update file, since it unzips to >2gb in addition to the almost 2gb already taken up if you have an almost full drive, or for some reason it otherwise can't, and then it throws random unrelated errors instead of just saying "hey we can't do this because you're out of space/the file can't be written" like it should. Pretty much just make sure you're not out of space and that you're using an administrator account on windows with no stupid restrictions on the size of my documents or anything. Also this is really dumb but, do you know anyone with/have access to a mac? I've instantly restored phones/other iOS devices that errored out over and over on a PC on one of my macs and can only really chalk it up to apple controlling the whole stack driver/software wise. It's also quite possible the phone was screwed up irreparably, at least with software and equipment you can use at home, by this update. I've heard about that happening occasionally since 8.0 came out. You might need to take it to an apple store or mail it in to apple to exchange. I've definitely told people to just go to the apple store with issues like this more than once.

Connect with USB cable to your PC running the latest version of iTunes. Your iPhone is almost full and doesn't' have enough space to download and unpack the update. So you need to do it trough iTunes on your computer. iTunes will download the update, back-up your iPhone, erase the phone, install the updated software and put your data back in place. If your iPhone is completely locked up you need to connect it to ihttps://support.apple.com/en-us/HT201252 Hopefully you have a recent back-up in iTunes and / or are you are using the iCloud back-up.

Not to threadsit, but just to clarify, it's a brand new phone so we're pretty sure that it can't be full, we have put it into iTunes to restore it and the problem that we're having is with the restore process on iTunes.

Did you purchase the phone online or at a store? Can you just return it?

Yep DFU mode and set-up as a new phone.

Update the laptop or home machine in every way before trying again. This has always been a problem when updating for me.

You might research the XP factor in this, too.

Thanks for your suggestions. I think we'll try going down the route of pre-downloading the IPSW so I'll post an update as to whether that worked later on. It's also good to know that this may well be a 'back to base' issue. We were starting to suspect as much.