Ooh, love this question! I always put a lot of thought into my ship designs and names.
My primary (VA Jack Harkness) always names his ships after things and people that were important to him. He now commands the U.S.S. Torchwood, but past ships included the Tyler, the Alonzo, etc. His first escort was the Garcon de Loyer (Rent Boy), because it seemed like a good name for an escort, and he thought it would get through the game's filters. (It did.)
My Caitian was a big, butch exchange student from the evil parallel universe, so of course his first ship was painted pink, and named the U.S.S. Sea Tiger, after the pink submarine in Operation Petticoat. He did not understand the irony of being given such a girly ship. He later had an escort named the Gaiman (after a favorite author), the Leviathan (because it looked a bit like a whale), and the Defiant-class U.S.S. Deviant (pronounced De-VI-ant, rhymes with Defiant). Oh, and then when he became RA1 and got his Prometheus class ship, he named it after me (@Propecius) sort of: U.S.S. Propetheus.
My liberated borg engineer has a Nebula class called the U.S.S. Nebulizer. His first ship was the USS Secondprize. His first science ship was the USS Britannic, after HMS Titanic's ill-fated sister ship, which served as a hospital ship for a while.
My science guy is naming all his ships after NOAA science vessels from the 20th and 21st century. Okeanos was his most recent, after the Okeanos Explorer.
Oh, and my Klingon (actually Orion)--who I don't play very much--has the IKS Barbie Fun Ship, again because I thought it would be a nice contrast to his gruff exterior. It's painted in pink and purple, as a ship with that name should be.
I could go on and on. But I won't. Obviously I love this topic!