And, of course, moments after I post the question I find the answer myself, it's a bug in 6.2. I need to update to latest version.

I chickened out and did not do the upgrade. I'm pretty new to DNN and never done an upgrade, and we're about to deliver, so... 

However, I did discover a very odd workaround. I navigate to the page with the apostrophe, then set its "include in menu" to the opposite of what it currently is, apply the change, and when the page updates and returns, presto, the Manage > Edit Content works! It works basically once. And you have to remember to undo the menu setting.

I don't remember where I saw the reference to it being fixed, but it was somewhere in the forums. My searches were always really simple like "apostrophe page name". The post mentioned the problem and said it would be fixed in the next release after 6.2.0, so they weren't saying it had already been fixed, just that it was coming.