Disable background app refresh for ONLY Messages app iOS 10 has made the Messages app drain my wife's iPhone 6's battery. 

It uses up about 40% of her battery and most of that is background time. 

How can I turn off background refresh for ONLY Messages?  

(Not looking for a solution that turns off all background refreshes.)
It sounds like you're trying to address the symptom, rather than the problem. To be clear, Messages should not be using up that much battery power. I suggest the following:


Restart your wife's iPhone by holding down the Power and Home buttons simultaneously for about 10 seconds until you see the Apple logo
Wait for her iPhone battery stats to refresh. 


NOTE:
This may take up to a day - typically it will need to be plugged back into a charger and fully charged and then removed and used normally for a few hours up to a day before the battery stats will refresh. 

Once you've done this, let us know if your wife is still having this problem.