dxkim said:
Wedge...it is possible that you have Wake On LAN set to active in your BIOS or a jumper on your Mobo. Consult your manual for confirmation.
As far as the WOL (wake on LAN). While your 'puter is in standby mode try to connect to it with a little file sharing from another computer (you said you were on a LAN...are there other computers on the LAN? you have file sharing enabled, yes?!). This should confirm if the WOL is enabled.
However, it could just be that your mouse is very sensitive. I used to have a Logitech ball mouse that was great for gaming and apps like Photoshop...but a door slam would cause enough of a vibration in the room to activate it and cause the system to wake from standby. My solution was when I walked away from my computer for a long time or purposely entered standby, I flipped the mouse upside-down. This causes the mouse ball to drop within the mouse itself and stay away from the internal rollers...thus no activation. 
As far as Internet security...you didn't mention if you are behind a hardware Firewall or not. Nevertheless, a Personal Firewall will protect you from the Net and others on the LAN. Might I suggest Kerio Personal Firewall as your choice...one of the best (if not the best) and its FREE!
h**p://www.kerio.com/us/kpf_home.html
I think the WOL feature isn't set to active because a message notifies me that I am "now connected to a LAN" whenever I come out of standby. i.e., I think it is waking everytime I come out of standby.
I'm sharing an internet connection with 3 other computers in the house. I have a cable router that connects them all and they each have complete and independent access to the net. The router itself does act as a hardware firewall, but I'm thinking of trying out the link to Kerio so thanks for that.
One other thing, I do not know how to go about sharing files among them. Is this even possible with the configuration I am talking about? It probably involves manually altering the router settings?
Another thing, my mouse is VERY sensitive as well. I have many times entered standby with my mouse only to reactivate the computer due to a tiny movement of the mouse when pulling my hand away from it. Good idea you had about turning your mouse over. In my case, I have an optical so that's a no go. But I have to admit, this is the first thing that came to my mind whenever my computer first did this. It could be the cause at some times? Who knows?
I'm beginning to think the problem is either the mouse (like you suggested), or that newby's suggestion about anti-virus software or other updating-type software being the cause.