I sometimes rename captured enemy
cities to fit in with the "flavor" or language of my own empire, but still bearing resemblance to their original names. For instance, when I play Korea, Japan or China I find it fun to mentally transcribe the captured city's name into my empire's language and then romanizing it back to alphabetic. For instance, Paris would become Ba Li, P'ari or Parisu.
As for
units, I love renaming highly (or somewhat) promoted ones
Woodsman 2 or 3: "Ranger"
Guerrilla 2 or 3: "Mountaineer"
City Raider 3+Barrage 1: "Citysmasher" - I may end up doing this for a lot of cannons!
Also Medic when appropriate. One really nifty one was a Gunship with major Medic upgrades. It was named "Medevac"

haha!
Edit: Ah. Pre-Marine era I might call my Rifleman a "Marine" if it has that same promotion. Or name a modern advanced infantry unit "Commando" or "Delta", if it's got the Commando promotion (use enemy roads).
Lastly. Cavalry units with both Flanking 2 and Mobility will be known as "Fast Cav" or something ^^