There is a new greasemonkey script made for Firefox that allows you to pick any past Google Doodle and make it permanent. The script can be located over here and like I said, requires Firefox and having Greasemonkey installed.
Once you install the script, you should go to Google.com and see the Google Lego logo. If you do not see the lego logo, then it was not installed correctly. Once installed properly, go to the Google Logo gallery and find the Doodle you like, then click on the “Make This My Favorite Doodle” button.
Whenever a new Google Doodle is shown, your favorite doodle will be replaced by the new doodle for that day. If you do not want the script to allow this, you can select “Favorite Doodle -> Always show my favorite doodle, even on holidays”.