IPhone – I cannot backup the iPhone 5

Try logging out and logging in again. There was an issue with some helper not restarting the right way after the required upgrades, relogging helped me. If that doesn't work, try rebooting.

Not all versions of iOS can back up from backups made using more recent versions (i.e. the backups are not necessarily forwards-compatible). Confusingly, rather than show these unusable backups in some way (e.g. disabled with an error message available on tap) they simply don't show up at all (at least with iOS 5.0.1).

A new-from-the-box iPhone isn't necessarily running the latest version of iOS; it's running what was latest (or close) when it was boxed. If the version on the phone is older than the version that was used to make the backup (which is actually quite likely now that iOS updates can be done over-the-air).

Thankfully, there is a simple fix: choose to set up the phone as a new phone, rather than from a backup. Skip the step where you are asked for an Apple ID, since you won't actually be using the phone like this.

Once the initial setup is done, plug the phone in to iTunes and choose "Restore" (you can skip the backup of the clean-state phone). This will download the latest version of iOS and install that first, and then offer to restore (either from backups on the computer or iCloud).