What I'm about to cover probably can't really be called a puzzle in the strictest sense, but they do make for fun thought exercises. The identity paradoxes in this puzzle guide are all related to the question of when something stops to be something. While they usually are paradoxes, you could arguably give an answer to each of them, so they are puzzles in that sense.
Paradoxes like this definitely aren't for every gaming group, but they could make for a fun and different story element. It could be a sly king testing a group of adventurers, a crazy god trying to mess with mortals, or a not too clever genie trying to trick people. The possibilities are endless in terms of story, but the paradoxes and their "solutions" will generally be the same, just merely told in different ways.
Let's start of simple, with something that isn't really a paradox, but merely questions identity. Let's say your brain is transplanted into the brainless body of somebody else in order to save you, similar to organ transplants, but on a far more advanced scale. Would you still be you? Would you be part you, and part whoever that other person was in life? Would you be mostly you, and only partially them? Or would you be mostly them? Since a brain makes up only about 2% of the weight of an average human body.
There are some clear answers you could give, each of which could be considered right. You could argue "you" would most definitely no longer be you, as the new body alone will cause all kinds of changes to your brain, this new body is definitely not like your old after all. But that does mean you're truly not you? Because in your current body you undergo all sorts of changes too. Are you no longer you when you're drunk, for example?
But let's take it a step further. Let's say your brain is put into a cloned body of you instead, one grown in a lab. Would you be you after the transplant? Everything certainly looks and feels the same. Let's even assume they've made sure your new body is an exact replica, meaning the same amount of muscle and fat tissue, the same scars, the same length of hair and nails, everything. Are you still you after the transplant? Now we're treading on paradox territory.
Now let's really push it into paradox territory. Let's say two completely identical people, grown in a lab for the sake of keeping them exactly identical, swap brains, are they still the same people? Or are they now different? They still have the same memories, the exact same bodies as before, and everything else is exactly the same, except that they had a brain transplant. "The transplant itself means a change was made, so therefore they're not the same as before." could be an answer, but does that really change who they are as a person? Besides, they both had the transplant, so even in that sense they are the sames. Plus if I were to cut my hair, I'd still be me even with that change, right? You could argue I wouldn't be, you could even argue we never stop changing, as every second something about us changes, but if I owed you something I wouldn't be able to say: "That was somebody else.".
Let's take a step back, and go in a different direction. Those familiar with the Ship of Theseus, or Theseus' Paradox, probably already saw this one coming, but the paradox, in altered form, is as follows:
Let's say I have a ship, and I replace the sails because the old ones were weathered and torn. Is it still the same ship? Now let's say I eventually end up replacing all the parts on my ship, but it still looks exactly like the ship I originally bought, is it still the same ship? If not, when did it stop being the same ship?
Alternatively, let's say I have a ship, took out parts and replaced them with new ones. I keep the old parts, and over time keep replacing the parts on my ship with new ones, while never throwing the old ones away. I eventually end up with all the original parts, and use it to build another ship. Which ship is the original one, and which is the copy? Or is neither a copy nor an original?
A different version of this paradox is Grandfather's Axe. It's the same idea, but in this there are only two parts that can be replaced: the handle, and the head. Is the axe, your trusty axe, still the same axe when the handle has been replaced to fix it? What about the head? Does it matter which part is replaced?
This could make for a fun or annoying quest, depending on how you experience it. A quest to return a sword, but the sword "isn't the same", because the pommel was replaced with a newer version of the old, and therefore the quest isn't completed, and the party doesn't deserve a reward.
Let's take another step back, way back, and look at some answers. This is still a guide for RPG puzzles after all, so we do want players to offer possible solutions to these paradoxes.
The solution to all of the previous paradoxes (and non-paradoxes) lies entirely within our language. The reason these paradoxes are paradoxes is because we don't have a concrete definition of what an original is and isn't. We solve this to an extent with objects, especially when those objects are rare. A vintage car that has had parts replaced won't be as valuable as a vintage car of the same make that still has all its original parts, for example. Things are trickier when it comes to people of course, as losing an arm wouldn't necessarily mean you're no longer you, it simply depends on the definition of "you".
So in short, you could consider the correct answer to all these paradox questions to be "It depends on definitions." or "These can be solved by inventing definitions." But if your players don't think of this, you could simply use any of their answers as a means to judge them (in character, of course). The character who posed the question could act in a specific way, or grant specific items or information depending on their answers.
There are plenty of variations you can play around with, some of which might fit your story universe better. Here are some examples you could play around with.
Say you have a heap of sand, and you take one grain of sand away. Is it still a heap? What if we continue to take away grains of sand one at a time, at one point does it stop being a heap? What even is a heap?
The reverse could also be asked. A character might have a single grain of sand in their hand, and ask: "Is this a heap of sand?". Then they will continue to add a single grain of sand one at a time, at what point does it become a heap?
The answer here is again dependent on a definition of a heap. Since there's no specified amount in a heap, it can be interpreted as all sorts of things.
A real life example is the difference between a mountain and a hill. At one point in time the (loose) definition of a mountain was anything above 1000ft./304m (based on the local area, not sea level), and anything below was a hill. This definition is no longer used though, but it shows how a heap of sand could stop being a heap when there's fewer than 300 grains of sand, for example.
If I take away a handful of sand from a beach, is that beach still the same? If not, that surely means the beach is never ever the same? It changes constantly after all.
If I took a glass worth of water out of a river, would that river still be the same? Yes? But if I have two glasses of water, they wouldn't be the same, right? But water constantly flows through a river, so a river is never the same, because the water is always different, or is it?
Let's say your consciousness, whatever that may be exactly, is uploaded into a computer, and you're in a simulation that allows you to continue to live as "you". Are you still you though?
Now what if a copy of you is made. That copy has the exact same memories, the same simulated body, everything's the same. Is that copy you?
To answer to this one could rely on whether or not the copy remembers the cloning process, and whether the process shows which one is the copy. If the process doesn't show this, neither one would know who the copy was, unless told by whoever did the copying. But even if the copy knows, does that mean they're not "you"?
Let's say a teleporter teleports you by scanning your entire being at a molecular level, then breaks you down, and builds you up again at a location of choice. Are you still you? Or did you die, and a new copy of you was "born" into existence?
This one is harder to give an answer to, even with definitions. But it could definitely make for a fun thought exercise, or at the very least a wacky character who's obsessed with how they have "died" over and over again, in a sci-fi setting.