My mini caught a bug or virus, the ended up with loginrescueme.com. So I did all I knew how to clean up my mini and keep her safe. Then this morning, I saw that in my System Preferences, Startup Disk, I now have an option of "Network Startup" that is

You're going to need to be more specific about what exactly is going on. Malware cannot install itself on your Mac, and the "logmeinrescue.com" site (assuming that's the one you meant... loginrescueme.com does not exist) is a legit site associated with the LogMeIn remote access software.

Did you see an alert pop up telling you that you have a virus, and prompting you to call a number, where someone directed you to a site where you downloaded LogMeIn and used it to give them remote access to your Mac? If so, you have been scammed, and your Mac should now be considered compromised. You will need to erase the hard drive, reinstall your apps from scratch and restore only documents (ie, no settings, apps, etc) from a backup.

If something different happened, please explain what that was.

Logmein Rescue is one of the modules available from Logmein for specialised usages (there's a conference module etc). It's legitimate but whoever was on the other end of it was most likely not.

If a 3rd party logged into your Mac consider it entirely compromised, in addition to Thomas's advice you should change any and all credit cards, bank accounts, purchase accounts etc, anything that is on your Mac that can be used to steal from you.

I can't answer that for you, I don't know what information was on your computer at the time, or what the interloper may have left behind for the future.

The choice of what to do must remain yours.

You're welcome.

So, just to be clear, you got a pop-up and called the number, and they directed you to install LogMeIn and then gained control of your Mac through that? If that's the case, your Mac cannot be considered safe to use. The people at that number are scammers, and may very well have done things to your Mac that no anti-virus software would ever detect and that a Mac security expert would be hard-pressed to find. (Keep in mind that the average Mac tech is quite far from being a security expert.)

Thus, you will need to erase the hard drive and reinstall everything from scratch. This must be done properly, as restoring the wrong things could restore something malicious.

For instructions on how to do this properly, see:

How to reinstall Mac OS X from scratch

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