Depending on the age of the mini and the OS it has, you might be looking at an impossible situation. The only solution would be going back to an older OS.

Older Mac models with newer OS releases and updates are becoming incapable of performing even the most basic of computing tasks such as surfing the web.

My 2004 PowerBook used to be plenty fast for all kinds of uses, but with the latest OS version and updates it can take, it can't realistically be used even when restricted to running only Safari. It's currently sporting the last version of Mac OS 10.5 but I should probably re-install the older 10.4. The big issue then is the older version of Safari or many other browser alternatives, won't work with some sites. No Google sites with the exception of search for starters. Since my wife uses GMail this would be an even bigger PITA than the constant freezing and sluggish behavior that exists now.