I don't like my ship name, but I can't seem to rename it. Is there somewhere in the inventory or journal?
I was just looking around trying to find out how to do this, and I found out how to do it by accident. When you're in your ship's inventory, if you look up to the top right there's an arrow in a box next to the ship's name. Just click it, it's that simple. I realize this thread is from last year but hopefully it still helps. If you didn't know there ya go!
At present, no, this is not possible. The only way to change the name of your ship is to change your actual ship (by purchase or by finding one on a planet) - and then your ship's name would change to the name of the new ship, although obviously you don't get to choose what the new name will be.
Sources: personal gameplay, that infamous Reddit post about all the stuff that was promised but not delivered in NMS.