My utility ships (colony, miner, contructor, scout, etc.) tend to get very utilitarian names, like "Colony Ship." I see no real reason to get fancy with ships of that type.
Warships, going from tiny up to medium hulls, get an interesting-sounding classification, but keep the names they are given at creation. For example, in my current game, my medium-hulled destroyers which I use to patrol my space are designated "Spartan class." Each individual ship just gets the name "Spartan" plus the number the computer assigns to it. My long-range attack cruisers, also medium-hulled, are the "Achilles class." These ships have an additional engine, as well as life-support systems, to facilitate excursions into enemy space. Because I have so many of these medium ships, giving each one a unique designation would be far too tedious. The same, obviously, goes for all of the small and tiny ships which are used to defend my colonies and fill out my fleets. My defense fighters are "Centurion class," and my attack fighters are "Hoplite class." The same naming scheme applies.
It isn't until I move up to the large and huge-hulled ships that unique names come into play. Because there tend to be far fewer of these ships, I can usually manage to name them individually, though sometimes it's just the huge ships that get this honor. My large-hulled battlecruisers use a naming system much like you would see in the U.S. Navy, or on Star Trek. The first ship of a class, the pathfinder, takes the name of that class, and each additional ship of that class takes a unique name. Currently, my large cruisers are the "Vigilant class," and other ships of that class are called things like "Intrepid," "Defiant," and "Vindicator." Very dramatic.
My huge-hulled ships, because they are so rare, usually get the same naming scheme from one game to another. As you may have noticed, I'm quite fond of Greek and Roman history and mythology, and I usually only build twelve huge ships at any given time, so some of you may see where this is going. Thus, my largest ships are the "Titan class," with individual ships taking the name of one of the twelve Titans of Greek myth: Cronus, Rhea, Hyperion, Oceanus, etc. So long as an enemy doesn't name one of its ships the "Zeus," I shouldn't have much to worry about.
All of my starbases tend to be named for their nearest star, and each one gets a corresponding rally point with the same name to facilitate upgrades. Military bases get the prefix "Starbase," econ bases get the prefix "Port," and influence bases get the prefix "Outpost." Bases that are located outside of my main space, far from a star system with an inhabitable planet, tend to get the prefix "Deep Space," followed by a number.
Lastly, upgraded versions of existing ships get the suffix "Mk" along with a number designating its generation.
And that's about it. Oh, how I wish some of this were automated. For example, starbases should automatically be given a corresponding rally point of the same name.