On a more philosophic basis the issue with the Ship of Theseus is that it presumes ships exist. They don't (no really). We call a collection of atoms a ship for convenience's sake but there's not actually a ship there, just a collection of parts that happen to move in the same direction due to physics. In the same way I can call something the "Mississippi River" and pretend it's the same river that Mark Twain steamed on even though not one drop of water is the same as it was then, the universe does not care that we happen to call one piece of flowing water the Mississipi and another the Nile, it's all hydrogen and oxygen atoms with some other assorted molecules in it. Insisting that the Ship of Theseus exists is like calling a character Bugs Bunny, and then arguing about which drawing in decades of cartoons is the real Bugs Bunny and which of those cells are not the real Bugs. They're all the real thing because Bugs Bunny is an idea that we're attributing some physical reality to even though the reality only exists in our own minds.
Consequently from that point of view it remains the Ship of Theseus so long as "Ship of Theseus" is printed on the bow, and becomes a different ship when those words are erased and "Ship of Hercules" is written there even if all of the parts are the same, because it's only the Ship of Theseus if we agree that this assortment of atoms is called the Ship of Theseus.
This argument doesn't really make sense to me, sorry. It might be getting philosophical, but I think you'd be nearer to the matter if you were going to argue about what makes a specific ship that specific ship. For example, I can make a new ship that is exactly the same as the ship that sailed in 1912, even making it out of the same type of inferior steel and iron, but the real RMS
Titanic is resting on the ocean floor, what's left of her. Likewise, people actually do argue about things like fictional characters being "real" or not. Saavik, for example (Kristy Alley vs. Robin Curtis). I even get into it myself with Kei and Yuri from
Dirty Pair, as I reject the versions from
Dirty Pair Flash. Kind of the same thing with nuTrek as far as rejecting the newer shows and movies. As for the Ship of Theseus argument, it's ironic in that I'm most familiar with it through the themes explored in
Ghost in the Shell and its various adaptations. How much of a human can be replaced and still be human? In the 1995 movie, only a few brain cells of the person Motoko Kusanagi was remained. In
Stand Alone Complex it was her entire brain. I'd actually argue that the brain is the only organ that can't be replaced while still remaining the same person.
Deep Space Nine actually explored that a bit, too, but there's real historical examples of changes in personality and the like resulting from injuries to the brain, like a guy who managed to survive being impaled through the head but was a different person afterwards. The other side of this question in GitS concerns AI: as AI becomes more and more advanced, it may become difficult to distinguish it from a living person. At what point does an AI become a person, if ever?