Simplicity is one of the most important qualities of Google's product. That's why I think Google's homepage should be extremely simple, without the options for image search, groups, maps or Froogle. You enter your query, press Google Search, and Google shows you the results and other services that may help you find the answer.
If you type "brazil photos", you are given the option to use Google Images and Google Maps. If you enter "shakira", you can view her profile on Google Music, videos on Google Video, buy her albums at Froogle and talk to her fans at Google Groups. If search for "george bush", you'll see Google News, Blog Search, Image Search and Google Video. For "binary search", Google Groups, Google Scholar and Google Books are helpful.
Services shown should depend on the query, so the homepage shouldn't display any service.
Another helpful addition might be shortcuts, similar to what Yahoo does. If you type "check mail", Gmail and Google Groups are relevant. Entering "dinner at the ritz Friday at 10PM", Google Calendar should help you add this event directly. If you say "search Gtalk tips in my bookmarks", an option to search in Google Bookmarks should be given. The ambiguity of searches like "mail Bill Gates" could be solved by showing each shortcut as an option.
A single search box can be more powerful than more disparate search boxes.
Related:
Google is a proxy
Design experiments in Google homepage and SERPs
