I've seen at least two people who have decided to name their ships "Icarus". I, myself, would be heartbroken if I found I could not name my ships that which I chose. I don't think there needs to be a 'uniqueness filter' for ships.
Second, I wholeheartedly agree with the idea of naming our ships. Not just to tell them apart in a shipyard, but that it gives you a more connected feeling to your ship. Yes, I am sentimental. To have someone target me and have both my name and my ship's name displayed in their target information would be amazing, like so:
Eagle - 'Talyn'
CMDR Talynne Star
Clean/Wanted
(Blah blah extra info)
Another option/addition, which I understand would be a bit more difficult, would be a decal on one side of your ship somewhere. When in outfitting, there could be a fourth Decal option. When you click on it, it would bring up a type box where you could enter your ship's name, as well as a drop-down tab with a few font selections...maybe five or six variations. I know that would be a lot of work in programming and such, but man it would make SUCH a difference, especially now with the external debug camera, to see your ship's name emblazoned on the side in the picture. I would be extra happy and proud to own that ship seeing that. ^_^