I use A Google A Day (http://www.agoogleaday.com/#game=started) site to help middle school students practice searching. It is a great practice tool because the searches change each day and no blog posts ruin your search practice. The questions are multileveled; students can't find the answer in one search. When students do this they must write down each query, the site they found, and what if any information they have found. This is very helpful if they are stuck, so I can see what they have tried. Also they write better queries, because they know I am going to check.
I have also used the Search Education site from google for lessons to help middle schoolers. I love the lessons!
It's not very user-friendly, but there is an 'art' of google searching. I remember some ad a while ago about a game or something by google to try to get better results. IIRC you play against someone else and whoever gets to a certain page or finds a certain result - or maybe it was point for result relevance - wins the round/game.
Also, A Google A Day. It give you a weird question that you can't just Google and have answered (it also uses a day old version of search results, as the exact question will likely turn up posts about AGAD). You have to figure out how to find the answer.
For example, I got "Who ran for the office of VP on the ticket with the candidate who lost the first time to the man honored at 200 Southeast 4th Street, Abilene, Kansas?"
So you'd have to figure out who is honored at that address, and when he won an election, and who ran against him in that election, and then who was on the ballot with that guy.
Oh yeah, here is one that I found a month ago on reddit. It is a really cool puzzle game where you have to find the answer through google. Pretty challenging and tests your googling skills.
I was initially under that impression too, but noticed they have a google.com e-mail address on their contact page, and appear to be searching historical Google indexes ("Deja Google," which appears to be an official Google project).
A bit of google-foo tenders this announcement on the official Google blog. It would appear the Google-a-Day project is officially a partnership with the NYT, although it is not clear (to me) which is writing the puzzles.
I love that site! Of course, the internet being what it is, by late morning people have already submitted the entire question, word-for-word, to Yahoo Answers or Quorra so you can, in fact, Google it, but....it's still fun.
It doesn't track the searches/sites that you use, but it does give you a timer.
one lazy Sunday afternoon of Autohotkey tutorial videos turns you into God at the office.
Also Google-fu training and challenges.
Stealing this! Featured snippets are the bane of my existence. I tried doing A Google A Day with my students during our research unit and all but about three of them threw complete hissy fits and slammed their computers shut with variations of "this is fucking stupid" when the very first snippet that popped up on Google wasn't the correct answer. Baby steps, I guess.
I remember this leaderboard for Xbox gamer points. Most likely one will go into some games blind. Kongregate has a “badge of the day” system, and http://www.agoogleaday.com has deja google to prevent spoilers
Personally, I like to refer people who seem to be unable to Google things to http://www.agoogleaday.com/ which is a fun daily game to help you learn how to find answers to various questions using Google.