Not Enough Space on iPhone to Restore Backup

I created a backup on my iPhone 5, reset the iPhone (Deleted all data), and am I trying to restore the iPhone from the backup. However, it won't work because:

iTunes could not restore the iPhone "My iPhone" because not enough free space is available on the iPhone.

Given that this was a backup from the same iPhone, there should be enough space to restore it.

More background information: when I did the backup, the storage space on the iPhone was almost full. Most of this storage space was used by what iTunes categorized as "Other"--space that stayed full despite deleting almost all data from the phone, including all photos, movies, Messages history, Safari history, etc. There were only a few apps on the phone, and nothing that would use much space. The Usage configuration pane didn't show any apps that were using this amount of space.

To try to resolve the problem, I took the backup and reset the phone. Now it won't even restore. Any suggestions on how to get the phone set up again, with all the (useful) app data that was there before, such as app data for game progress?

Thanks!