Updating my original list with a few more things to try if you're having consistent problems making a backup.
One problem that is probably well addressed in this thread is a broken previous backup. In those cases deleting (or archiving, so you don't lose it) the previous backup will push the issue through. This is the most frequent cause I see for the problem.https://support.apple.com/en-us/HT203282 (Just the section entitled "Archive an iOS backup with iTunes")
Another common problem is a flood of notifications during the backup causes the backup process to need to restart enough times that it eventually just crashes the backup program. You'll frequently see this one produce a message about the iPhone disconnecting. You can try putting the device into Airplane Mode in order to test/prevent this cause.
In Kiwimacguy's case, there's an intermittent problem using the iPhone's keychain, that causes the backup to fail sometimes and succeed other times. At some point, changing the lock screen password just wreaked havoc that only a backup restore could fix, but the root cause definitely seems hardware related. This is likely a more rare cause of this error message.
On Windows computers, there seems to be an increase in USB connection problems beyond just a faulty cable (although cable problems are frequent too). I haven't quite put my finger on the root of this issue enough to post here yet, but I've been seeing it happen a lot more over the past few weeks. If you haven't tried already, try switching out the cable, switching USB ports, rebooting the iPhone, and resetting the Windows sockets ("netsh winsock reset" in an admistrator command prompt.)
Another way to resolve some USB or wifi connection problems that may mess up the backup is to Reset Network Settings from Settings -> General -> Reset.
Several people have reported incorrectly installed apps causing their backups to consistently fail. In this case, open the Settings app -> General -> Usage -> Manage Storage and look through your installed app usage. (It may take a moment for the list of apps to appear.) Look for apps named "Null" or with no name. These apps need to be removed for the backup to proceed. Here are two other forum posts about this specific issue. "Null" app can't be deleted and Backup fails at step 4