According to item 4 of this Google seminar writeup nobody uses the "I'm feeling lucky" button. Must be because they don't know how. "I'm feeling lucky" is absolute essential to navigate websites with crummy site navigation and/or search facilities.
What you do is create an "I'm feeling lucky" search shortcut (in mozilla firefox obviously - you have switched haven't you?) using the site:crummysearch.org search parameter (or alternatively the inurl parameter) to restrict to the poor site in question. Since Pagerank works so much better than most website ranking algorithms this actally works much better than the search page on the website itself. You usually get there first go. Good cases where it works are (at least) wikipedia, IMDB and allmusic.com.
Your average website (including this one) has surprisingly crummy search facilities. It's one thing that they're slow - but the second thing is that people have typically done a premature optimization and indexed the content in the database instead of indexing the shown pages. For this weblog, for instance, that means that page comments don't get indexed along with their posts. This is a very common phenomenon. It is particularly annoying when people misunderstand the web and do a full robot block via robots.txt (e.g. all Danish newspapers do this). Their own search facilities are no match to Google's - which means their sites have about 1/10th the usability they could have if they were Google indexed.
Meanwhile, we're waiting for "Google Tags" - useful stored public searches, available at urls like this: http://tags.google.com/myspecifictag - Maybe I should just make a service like that here on classy.dk. Obviously, there are resources like the The Google Hacking Database, but they don't have live queries...