In most games where I have to name ships (there's not many, unfortunately) I name them things like Intrepid, Defiant, and such...but since most of those ideas came from Trek, I want to use something different. I just hit the random name button until it comes up with something I both like and sounds semi-uniqueish. Some pretty random stuff generally comes up, but my current main is the USS Grim (which doesn't sound like a typical TV Trek Starfleet name, but goes good for a hybrid Prometheus escort).
I do the same thing with characters, since I never play the big three races (humans, vulcans, and klingons), and I don't know what the naming traditions are for others. I'm pretty sure my KDF Alien is just some random letters.