I envisage a system where the game offers to rename a unit once it reaches a certain experience level. When a unit gets to this level, the rename pop-up comes up and the player can manually rename the unit, or hit cancel.
Ideally, the threshold for this rename should be user-definable in the BUG options screen. This it to maintain suitability with a variety of mods. For example, level 6 would be suitable for otherwise-vanilla Civ, but would be too low for Rise of Mankind (where various civic, building and great general bonuses mean that a unit can actually start out at level 6).
This would help the player to keep track of their most experienced units, and introduce an almost 'rpg' feel to units.