How to change the location where iTunes stores backups of iPhones?
For Mac OS X, I had to use this guide: http://osxdaily.com/2011/03/05/how-to-move-the-iphoneipad-itunes-backup-folder-to-an-external-hard-drive/ and create a symbolic link to move the Backup folder, but it worked great.

The main thing is to move the Backup folder to an external drive/partition that has more space and then create a symbolic link like so:

ln -s /Volumes/YourExternalDrive/Backup/ ~/Library/Application Support/MobileSync/Backup


Then execute a Backup by syncing via iTunes to try it out.