It is kinda hard to suggest Software features when Hardware dont improve. Adding more things takes more Memory, CPU resources etc....
Most of the things I have problems with are iCloud related. Like Voice recognition which sometimes doesn't work, i have to repeatedly say the same thing and it doesn't connect. So it is not a recognition problem, but connection problem.
App Store, better search and suggestions. Currently it sucks after all this years. ( Again iCloud Related )
WiFi preload, try to preload most of the information when I am connected to a WiFi and charging my iPhone. For example, When you go back to office ( WiFi ) and Charging, iPhone will preload most of the information in background, and try to pull as much information as possible when you unplug the iPhone. Apps like Facebook, Twitter, Bloomberg, or other most often used news serveries. The idea is when you go out for lunch you will likely have all the update already. This will dramatically save Mobile Network data usage.
Option to make Voice Call an Notification instead of full Screen. Do anyone make calls anymore? Rarely, at least according to stats. So why should a call distrub me when I am on full screen movie or playing a video game? Voice Call should be like any other messaging app, just a simple notification will do.
I really wish Visual VoiceMail could be a standard and free. I can pick up the voicemail when ever i want to.
Settings are in need of a DEEP rethink.
Use Apple Music as Ringtone. ( You can customize which part of the 30 seconds to ring ), Again an Services thing rather then iOS 10.
Some Photos should be hidden by default. These days it is far too common people asking to see others photo album, and there are things you dont want them to see.....
Get rid of some Default apps, or simply make them great. Weather needs their local notification, mssing a lot of these government weather warnings from countries.
Tab for Phonebook? I have been asked far too many times how to make the Phonebook to show clients ( for Work ) and Real life Friends only.
Get rid of 16GB - ( For Pete Sake this is 2016 )
iPhone upgrade program should include iPhone capacity iCloud Storage space for Backup.
Scrolling in "peek" views. - ( No need to hold on to it to preview, one more tap to pop. )
Video Viewing, can the God Damn Quicktime player just at least support MKV. ( Sames goes to OSX as well )
Need a little more Visual Clue, current design trend is stupid. Taking away too many things.
Larger dictation button on keyboard, or some shortcut for it to activate.
There is quite of lot of under the hood changes, things like fully utilize Metal top to bottom.
Go 64Bit, ARMv8 only.
The BLEEMING Alarm clock needs to learn from my Calendar When is PUBLIC HOLIDAY.
By the way Calander should default give you all the public holiday for your location / region.