Nice thing about the Dalai Lama example is that he probably gets strong social pressure to believe he is a continuation of his previous incarnation. I don't know about the details of his office, but if I really believed in reincarnation, I would prepare a lot of notes for my future incarnation... something like what every reasonable person would prepare for themselves if they expected to have a sudden memory loss at some moment in the near future, assuming they would want to follow their original plans. (Perhaps how much of this Dalai Lama really does could be used as a measure of how much he literally believes in his reincarnation.)
There are also social pressures in the opposite direction, just smaller ones. People are supposed to "change" during various rituals, not necessarily religious ones: finishing a university and having a title added to your name is a secular example. In a work or in military, when you change your position in the hierarchy, it's not only about what you do, but also how you behave towards others, and how the others behave towards you: so it's like a minor surgery to your personality.
I am curious whether (as a part of a mad science experiment) it would be possible to create an opposite of the Dalai Lama effect; to create a Monday Man who would believe he is a different person that the Sunday Man, despite having the same body. More precisely, how far you could get, using only beliefs and social pressure on a neurotypical person.
To some degree this experiment was already done, by various cults. The cultists are supposed to believe they are someone more or less different than they were before; they even use "born again" to describe the change. But it seems like the old personality continues to exist -- at least some descriptions of "deprogramming" claim that if a cultist is kidnapped, prevented contact with other cultists, and prevented from doing their mental rituals (e.g. if you remove all the external and internal pressure towards the cult) that is usually enough for the old personality to reappear. If this is true, I would consider this an unsuccessful change. A successful change would be where the new person is completely free to do whatever they want, and they still naturally remain the new person. Even better, if the Sunday Man could regularly become a Monday Man every week, and then regularly change back again, both personalities having their own lives.
One obvious problem is memory. It stays with the body. The Dalai Lama can partially copy it from the old body to the new body, using notes. But you cannot keep two separate instances for the Sunday Man and Monday Man. Could people believe (if this is what their culture would tell them) that they can have a memory of another person in their head, but it doesn't make them identical with the person? Would the Monday Man accept that he has Sunday Man's memories, but he is a different person? The environment could be different, e.g. the Monday Man would live in a different house, wear a different uniform, meet different people, and the people would behave differently...
In other words, if you are not insane, you are probably not experimenting enough.
Something like the Monday Man already exists. In African and Afro-American religions such as Voodoo and Candomblé, people who get possessed say that whatever was moving their body in that time was not them, but some other entity. They frequently claim amnesia about the event, saying their normal non-possessed selves were "not there" to even notice what was happening. I don't know if anyone has done experiments to see whether they actually lack access to memories of that period, or whether they're merely denying them.
Of course, these religions, and particularly possession states, often involve great amounts of strong alcohol, so maybe the amnesia thing isn't so far-fetched.
This recent SMBC comic illustrates the old question of what exactly is you by referencing the Star Trek Teleporter Problem. Do you actually get teleported or does the teleporter just kill you before making a copy of you somewhere else?
Well, the answer that a lot of rationalist seem to accept is Pattern Identity Theory proposed by Hans Moravec (skim the link or do a google search for the theory if you have no idea what I am referring to). I am very sympathetic to this view and it definitely ties with my limited understanding of physics and biology - elementary particles are interchangeable and do not have 'identity', at least some of the atoms in your body (including some of those who form neurons) get replaced over time etc.
This is all fine and dandy, but if you take this view to its logical extreme it looks like a sufficently modified version of you shouldn't actually qualify as you - the difference in the pattern might be as great or greater than the difference in the patterns of any two random people.
Let's say something happens to Eliezer and he gets successfully cryo-preserved in 2014. Then 80 years later the singularity hasn't arrived yet but the future is still pretty good - everyone is smart and happy due to enhancements, ageing is a thing of the past and we have the technology to wake cryopreserved people up. The people in that future build Eliezer a new body, restore the information from his brain and apply all the standard enhancements on him and then they wake him up. The person who wakes up remembers all that good old Eliezer did and seems to act like you would expect an enhanced Eliezer to act. However, if you examine things closely the difference between 2014!Eliezer and 2094!Eliezer is actually bigger than the difference between 2014!Eliezer and let's say 2014!Yvain due to having all the new standard enhancements. Does that person really qualify as the same person according to Pattern Identity Theory, then? Sure, he originates from Eliezer and arguably the difference between the two is similar to the difference between kid!Eliezer and adult!Eliezer but is it really the same pattern? If you believe that you really are the pattern then how can you not think of Eliezer!2014 as a dead man?
Sure, you could argue that continual change (as opposed to the sudden change in the cryo!Eliezer scenario) or 'evolution of the pattern' is in some way relevant but why would that be? The only somewhat reasonable argument for that I've seen is 'because it looks like this is what I care about'. That's fine with me but my personal preference is closer to 'I want to continue existing and experiencing things'; I don't care if anything that looks like me or thinks it's me is experiencing stuff - I want me (whatever that is) to continue living and doing stuff. And so far it looks really plausible that me is the pattern which sadly leaves me to think that maybe changing the pattern is a bad idea.
I know that this line of thinking can damn you to eternal stagnation but it seems worth exploring before teleporters, uploading, big self-enhancements etc. come along which is why I am starting this discussion. Additionally, a part of the problem might be that there is some confusion about definitions going on but I'd like to see where. Furthermore, 'the difference in the pattern' seems both somehow hard to quantify and more importantly - it doesn't look like something that could have a clear cut-off as in 'if the pattern differs by more than 10% you are a different person'. At any rate, whatever that cut-off is, it still seems pretty clear that tenoke!2000 differs enough from me to be considered dead.
As an exercise at home I will leave you to think about what this whole line of thinking implies if you combine it with MWII-style quantum immortality.