Solved!
You can do this from the phone, or from your PC using web browser.
PC/Browser: https://www.google.com/history
Log in using the same google account that your droid does. You can manage, and even disable the 'feature' to save your searches here.
From the phone:
setting->general->search->Google Search settings->Manage Search History
Once you've hit this (Manage Search History), it opens a browser, and you manage the 'feature' the same way you would if doing it from PC/Browser.
Hint: Use the Remove Items menu to clear - link to Clear all History will pause the 'service'.