I like the idea.
It could store all the user created names in a file somewhere. Each time you name a ship, if it's a new unique name it gets added. When you click to rename, you can use up/down arrows to select names from this list. A popup will be trickier because if the list of names grows large enough (and on smaller game resolutions) it'll run out of room, and there's no scrolling functionality built into the UI windows in Sins.
It could be as simple as just dumping all of them unorganized, or it could be organized by race or even ship type (so if you're renaming a Kol it would only show you names you've given to Kols in the past, etc).
The file could probably be dumped in the Setting folder where achievements and keymapping go.