There was absolutely no system used to name merchant ships at this time — the owner would select a name that he liked, or if purchasing an already-named ship, would generally keep its name (sailors consider renaming ships to be Very Bad Luck). There was no such thing as a central registry of ships at the time, except inasmuch as maritime insurers kept records of deeds, which in any case would include he ship’s name, the insured owner and the ship’s master or captain.

Within navies, ship naming was somewhat more regulated in the sense that ships often took the names of precious ships — the Victory of 1754 that you can see at Portsmouth traces its name to Sir John Hawkins’ flagship in 1588; and multiple RN ships and shore facilities have been named for the French Temeraire captured at the battle of Lagos in 1759. But even at Trafalgar, there were three ships named Neptune or Neptuno, belonging to each of the participating navies.