I think it would be really interesting to see how much damage you are able to deal with a champion at all stages of the game. Take teemo for example, I want to know it what matches the majority of my damage came from shrooms (aka ambient damage), or single target damage. It would really help me to determine how i should build given past circumstances. Obviously the mechanism to create a damage metre is there as all damage is shown to us at the end of the match, and to a lesser degree on death recaps.
For instance I just had a match where I went tankmo, delaying my core build by building a thornmail and a sunfire first, Im really uncertain how this allowed me to deal 66k damage in a 35-40 min match but a damage meter would actually allow me to see exactly when and where my damage output spiked, or if it was consistent throughout the game.