Thanks, again, for the quick responses. I have tried two different monitors on my mini, one our newer 17" flat screen and the other an older 17" CRT monitor. I have not been able to get it to come on at all today. This has always happened to us while we were using it. This is the exact same pattern as before the motherboard replacement. Another thing they did at CompUSA was to reset the PMU before they replaced the motherboard, to no avail. I am going to try to talk to the repair tech on Monday and see if he even tested the power brick. There is no mention of them trying a working one, which is what I wanted them to do first.
But now that it won't come on at all, again, I guess we will be taking the trip and we will be without a system for at least another two weeks. We have had Macs for 13 years now, and have never experienced anything like this. Maybe Apple needs to bring some of their retired engineers out of retirement.