Any more info you can provide? What Mac OS X version do you use and what does that restart entail? Do you work on something with the Mac mini and suddenly the Mac mini just restarts and does not give you any info? Maybe check the Console for hints.
Have you already tinkered with its hardware as you wanted to?
"How to maximise your MacRumors troubleshooting experience" created by mad jew in 2006
When I am watching videos?
I guess we have to pull everything out of your nose then and this thread will have many posts.
What videos?
Online or via a video player like QuickTime or VLC Player?
And again, does the Mac just shut off and then restart and log back into your account with no message window at all? Have you checked Console yet?
And do you know the Mac OS X version you run?
AND PLEASE, BEFORE YOU ANSWER ONLY ONE QUESTION AGAIN:
"How to maximise your MacRumors troubleshooting experience" created by mad jew in 2006
Online-videos
Osx- Snow Leopard
The computer just restarts and goes to desktop
Does it also happen in another, even temporarily created, user account?
Since when does it happen, is everything up to date (Software Update) including browser and Flash (which you probably use?)?
Have you tried deleting "com.apple.Safari.plist" in Macintosh HD / Users / YOU / Library / Preferences / yet, guessing you using Safari?
And I guess Console is a no go too?