This is a bit of a thread hijack, but I feel a certain amount of the preceding material is necessary for the discussion. In a new thread you'd just have similar people hashing out similar things again before the topic got started anyway.
We actually discussed this quite a bit
here. In fact, that thread seems to be an appropriate place to move the discussion if you want to hash it out.
A common convention which works 90% of the time is material continuity.
Agree that this doesn't work.
Another convention, my preferred one, is pattern continuity. This certainly has problems, as our minds are stopped
I'm not sure what you mean by pattern continuity here; based on the problem you cite, it seems you're holding a sort of axiom that there should be some "always on" computer, and if it's ever shut off, the pattern ends, and any reboot is a different thing altogether. Given this description, I don't think this works either.
Yet a third, which it seems you switched to when ditching the first, is... I don't know what to call it. But as long as there's still someone in the universe who can legitimately call himself you (by some standard, opinions wildly differ here), what happens to you you is inconsequential.
This one seems to assume that somehow you can be in two places at once, and I don't think it works for that reason.
Given at least my interpretation, I don't like either of the above--no wonder you find them irrational!
I propose a fourth alternative--informational continuity. Specifically, there are certain kinds of things that only I am privy to, and I can develop memories of these sorts of things. In situations where those sorts of memories are genuine, I should have all rights to claim that I remember being that person.
I think this is the critical invariant for continuity, and it dodges all three of your problems. For the Ship of Theseus issue, it really doesn't matter what molecules are in my body--it matters more what the mental states represent, and whether the represented states have a "correct" causal relationship to what they represent. As for the pattern issue, so long as the information got carried, it's not a problem if everything stopped--if you were frozen in carbonite and reconstituted, you still have rights to claim to be the same person.
The cloning part of the infamous teleporter is the "tricky" one, but it's not really too bad if you think about it. This makes two bona fide separate individuals; each should care about the other just as they would their twin. Neither can claim to be the other, because neither is causally linked to the other's subjectivity in the right way. However, both can claim to have been the original, because they were both causally related in the appropriate way to their past counterparts.