Thought experiment: Suppose Satoshi, whoever it is, is still alive somewhere in the wild. But he doesn't have his original keys (hard-drive failure with no backup, apartment fire with no offsite backup, etc etc).
He could still have access to his PGP private key, which would be a strong data point. It could be possible that someone stole this key though, so this alone wouldn't completely prove his identity.
If he had the foresight to know he might want to prove his identity at some point in the future, he could have embedded the hash of a message into an early block (using the public keys of coinbase transactions). The message could say something like "Satoshi Nakamoto is <real name>. <salt>". In order to prove his identity, he'd just have to reveal the contents of the message (he'd have the make the salt something he could memorize so that it wouldn't be at risk of being destroyed by a fire). Ideally he'd encode this message hash in multiple early blocks (using a different salt in each message, so that the hash is different each time).
I think anything like the "knowledge of the content of private emails" is flawed. Satoshi's email account was hacked, so most of his "private" emails are no longer private (supposedly he didn't use PGP for most of his private correspondence).
A Bitcoin private key is only 51 characters when expressed in the Base58 format that Bitcoin uses. Here is an example private key expressed in such a format: 5Kb8kLf9zgWQnogidDA76MzPL6TsZZY36hWXMssSzNydYXYB9KF
If I were Satoshi I totally would've committed it to memory. I have more random characters than that memorized for passwords to various accounts.
Or, even easier, have a smaller simple password, and then be able to reconstruct them using an algorithm that your remember, e.g. SHA256(SHA256(SHA256("correct") + "horse battery") + "staple").
According to that awesome analysis[0], he was backing his stuff up pretty regularly (every five days at first). I would find it very hard to believe that someone with the wherewithal to make backups is not making an off-site copy of those backups. I haven't met many people that make backups but only store a copy locally. History is rife with horror stories. It's a good question though. Specifically there might be things only Satoshi knows among some of the original adopters (but clearly Gavin is not one of those people, if he was fooled by Crazy Craig).
I believe that David Kleiman was the one with the Satoshi keys, they were kept on an encrypted USB. He died unexpectedly in 2013, and his brother took the USB. Nobody knows how to unlock it.
How would Satoshi prove himself then?