Irrelevant information is just information that doesn't change the probabilities. If this does, it's relevant.
Irrelevant information is just information that doesn't change the probabilities as long as you follow the axioms of probability. If we speculate that always assigning 50% to heads can be the wrong method, i.e. axiom-violating, then deciding which information is relevant based on it is putting the cart before the horse.
What other ways can you tell whether information is relevant? Bayes' rule is a good tool for it, because you know it follows the right axioms of probability. Here is the path I followed: If the probabilities of the two "worlds" are 1/3 and 2/3, you expect to see 50% heads and 50% tails on the second coin. If the probabilities in the two worlds are 1/2 and 1/2, you still expect 50/50. The probabilities on the second coin are then 50/50 no matter which rule is right. If see heads or tails, then Bayes' rule says we should update our probabilities by a factor of P(B|A)/P(B), or 0.5/0.5, or 1. No change. Since we trust Bayes' rule to follow the axioms of probability, something that disagrees with it doesn't.
Or you might go the conservation of expected evidence route. If the second coin landing heads makes you change your probabilities one way, conservation of expected evidence (another thing that has a fairly short and trustworthy derivation from the axioms of probability) says that the coin landing tails should make you change your probabilities the opposite way. Does it?
The underlying "why" reason the information is irrelevant is because in our causal world, you don't get a correlation (i.e. information about one from knowing the other) without having a causal path between the two events - like a common ancestor, or conditioning on a common causal descendant. But the coinflips were independent when flipped, and we didn't condition on any causal descendants of the second coinflip (like, say, creating more copies).
leeping Beauty is put to sleep on Sunday. If the coin lands on heads, she is awakened only on Monday. If it lands on tails, she is awaken on Monday and Tuesday, and has her memory erased between them. Each time she is awoken, she is asked how likely it is the coin landed on tails.
According to the one theory, she would figure it's twice as likely to be her if the coin landed on tails, so it's now twice as likely to be tales. According to another, she would figure that the world she's in isn't eliminated by heads or tails, so it's equally likely. I'd like to use the second possibility, and add a simple modification:
The coin is tossed a second time. She's shown the result of this toss on Monday, and the opposite on Tuesday (if she's awake for it). She wakes up, and believes that there are four equally probable results: HH, HT, TH, and TT. She then is shown heads. This will happen at some point unless the coin has the result HT. In that case, she is only woken once, and is shown tails. She now spreads the probability between the remaining three outcomes: HH, TH, and TT. She is asked how likely it is that the coin landed on heads. She gives 1/3. Thanks to this modification, she got the same answer as if she had used SIA.
Now suppose that, instead of being told the result of second coin toss, she had some other observation. Perhaps she observed how tired she was when she woke up, or how long it took to open her eyes, or something else. In any case, if it's an unlikely observation, it probably won't happen twice, so she's about twice as likely to make it if she wakes up twice.
Edit: SIA and SSA don't seem to be what I thought they were. In both cases, you get approximately 1/3. As far as I can figure, the reason Wikipedia states that you get 1/2 with SIA is that it uses sleeping beauty during the course of this experiment as the entire reference class (rather than all existent observers). I've seen someone use this logic before (they only updated on the existence of such an observer). Does anyone know what it's called?