Sorry to dig up this old thread but I just had the same problem as the original poster. Firstly, I'd like to say that there were a lot of very unhelpful posts both in this forum and in other places. I eventually managed to do almost exactly what the OP was trying to do.
I started with a backup from a 128GB iPhone 6 and restored it to a 32GB iphone 5S. The backup was too large for the 32GB iPhone, so I kept whenever I tried the standard "Restore Backup" option, I got an error saying there was not enough space on the device. I didn't mind if I had to get rid of some apps, music, pictures, etc, but there was not even an option to do that. It would just fail, then all I could do was click on "Ok" and the phone would restart. I was trying not to update the iOS version, because I wanted to stay on 9.0.2.
Anyway, eventually I realised that wasn't going to be an option. I put my iPhone 5S into recovery mode and did a full restore/update. Now this time when I chose to restore from my large iPhone 6 backup, it actually let me do it! It started populating the phone with my data: "Restore in Progress". Eventually the iPhone was back on the Springboard screen and the apps were slowly being installed. At this stage, I was able to deselect a whole lot of music and photos and even delete unnecessary apps - either from the phone itself, or from iTunes. So now I have managed to squeeze my most important apps and data onto this 32GB iPhone 5S, from what was originally most likely a 80GB+ backup.
I hope this helps someone else in the future.
[Edit: iBackupBot may be another solution, but it is not free]
Sorry to reanimate an old thread but I've tried this solution a couple of times to try and get one of two 5S 32GB backups (12.8 GB & 11.8 GB) to my new 6S 16GB with no luck.
A couple of questions:
Has this 'option' been removed in the latest version?
If there's a backup from a higher capacity iPhone (32GB in this case), then what size does the backup need to be in order to fit on a lower capacity (16GB) iPhone?
Any help would be appreciated
(PS. This was a replacement from my insurance company and I had no option to get a higher capacity phone, only take the cash value of my old phone and start again)