Prioritize the data that really matters
No one wants to lose their contacts and photos when their 6s falls in the toilet, but does it really matter if Words With Friends and Tinder are restored on your new device? Hopefully not. By default, these apps automatically back up to the iCloud, so you can free up a decent amount of space by disabling that option.
Go to Settings > General > Storage & iCloud Usage > tap Manage Storage under the iCloud section > select your device ("This iPhone") > tap Show All Apps. Now, go through and toggle off all the apps you don't need to do a restore (bye bye, Snapchat). Seeing how much space each takes up should help motivate you.
You'll notice that "Photo Library" is by far taking up the most space. Turn that off, too. Don't freak out, this won't delete any of your photos (and you just transferred them to your computer, remember?) "But how will I back up my selfies!?" you protest. Keep reading.