We've all been down that road and it's clear that most people do not like the name "Nottingham East Midlands Airport". I certainly don't. I think the name, apart from sounding horrific, actually sort of restricts the catchment area even more as the airport also serves Leicestershire, Derbyshire, Lincolnshire and South Yorkshire as well as parts of Northamptonshire. East Midlands is a as well used by us in Leicester and those in Derby as by those going to/from Nottingham. I think the current name "East Midlands Airport, Nottingham Leicester Derby" is a fine name and actually gives an even better idea of where the airport is located than just using Nottingham.
I used to hate the days where on the Ryanair website I had to scoll down to find "Nottingham East Midlands". The fact remains, the local people are always going to simply refer to it as East Midlands airport so having the cities included as a sort of extention to the name is far better than just adding one of the major cities nearby in the main name.