I did not read through this thread so I am sure I am putting some ideas that are out there already.
With that disclaimer out of the way, here we go...
-Cards.
By this I mean a new way to envision the multi-tasking system used in iOS. I would like it if you could swipe up from the bottom of the screen and then access a set of cards which you can then see what the app’s state is (visuals are always a nice thing) then have the ability to just swipe them away a-la webOS. Heck others have “borrowed” from webOS time for Apple to take a look as well.
They could also use this area for quick toggles for Wi-Fi, Airplane mode, Bluetooth, etc…
-Actionable notifications would be a great thing as well. Just click on the notification and get on with your business.
-Three finger pinch on screen to exit out of apps rather than the home button.
-Make the weather icon dynamic.
-Better Siri integration locally on the device.
-Oh and though this falls under hardware, the home button be it the way it is now or capacitive, have a ring around the home button that glows different colours depending on the notification you missed.
That should be good for a start.