I rename cities a turn after they are created. That way the default name is removed from the naming queue since it has already been used. However, I mostly only rename ones where the name is difficult for me to identify easily, for instance about half the Aztec city names look the same to me, so I change them. Mostly, I modify the name to keep its flavor but make it stand out in my mind. Occasionally, I will run a made up empire name and change every city name as soon as it is created, since I won't be using any of the default ones.
As for units, I only name the ones where I am sending the same basic type down different promotion paths. Then I name all the ones from each path the same. That makes it easy to choose exactly the type unit I want for any battle without having to look over its promotions. For example, my most common use of this is with tanks and advanced armor. I promote some of them with City Raider promotions first and rename them as "Raider", while a second group, intended for stack defense and attacks on enemy units in the field, gets the basic general purpose promotions and renamed as "Battle".