I imagine that the brand won't have much mindshare after another 10 or so years. It is already pretty much dead to me other than a few email accounts. After it tanks, it will probably get picked up by someone and people will continue to be able to access their Yahoo mail.

Remember Webcrawler? That was the tool to use prior to Google, for a short amount of time at least. It's still there http://www.webcrawler.com/ and owned by Infospace. Is it relevant? No. But it still exists.

Eventually those that lose to greater companies (like Google) on the web will totally fade away, but unlike old department stores of the past who haven't left a much of a trace in their wake, it is much easier to buy old web assets up and continue hosting them, if for no other reason than to steal their traffic for marketing purposes.