|laziness, impatience, and hubris|
First off, the standard disclaimer: IANAL.
Fair use also contains provisions for copying information that you can't get by any other means. It's a tricky area, because obviously you're not allowed to copy and sell out-of-print books. I'm pretty sure fair use covers copying an out-of-print book rather than buying it, though, since there's no practical way to acquire that book except by making a copy. By contrast, copying a book that's in print rather than buying it would be a clear violation of copyright.
With regard to the original question, I'd say your best bet is to start by trying to contact the original creator, and ask permission to mirror/redistribute the work. If you do make contact, and do get permission, you're free and clear, and we don't have to talk about fair use at all.
If you can't contact the original creator, your safest bet is to link to the existing site from your own. Use the original as an historical reference, and build your own body of new concent around that. Make a personal-use copy that can serve as a mirror in case the existing site goes away, but only put that mirror online if the existing site does go away.
If you do put up a mirror of no-longer-available content, put a big disclaimer at the top of the document tree citing the original creator and listing the URL where the information used to live. Then add a 'statement of good faith', saying you want to contact the original creator and get permission to mirror, but haven't been able to do so yet. Also announce that you will pull the information offline if the original creator contacts you and tells you to pull the plug. Then post your own contact information, so the original creator can reach you if he ever stumbles across your mirror.
That approach should keep you safe. In cases where people can't get the information any other way, and you've tried to contact the original creator but failed, the courts will probably allow you to run the mirror in good faith (i.e.: assuming you would have gotten permisssion to mirror) until the original creator says otherwise.
If you really care about the information, don't just redistribute 'dead' documents. Observe the spirit of the Open Source/Free Software movements, and do something with the information. Use what's there as a foundation for something new. The more of your own sweat you put into a project, the more it becomes your own, both emotionally and legally. Build clean versions of the original pages, based on your own knowledge and research. Add new information. Patch holes in what's there. Cite additional sources for the same information. Get other people talking about the subject, and post those conversations with permission.
Then make damn sure you've given others permission to build upon your work.
The best way to preserve information is to keep it alive. Maintaining a dead site is like tending a mummy. With appropriate care, you can make them last a long time, but at the end of the day, they just sit there being dead at you. Tending living information is like raising a family. The bloodline will go on indefinitely, even though individual members pass away.