Mac Mini boots and runs very slo-o-o-wly

Question

I bought my Mom a Mac mini with Intel Core Solo processor. I live in another city, and was last in front of the computer in February. I came for a visit yesterday and last night noticed the screen seemed to be frozen on the screen saver. I turned the monitor off and went to bed. When I came to look at the computer this morning it was frozen at a Mozilla Firefox upgrade install. I learned from my Mom that my sister, who is not particularly computer literate, used the computer last, about two weeks ago, to try and download some pictures. I'm writing for help, using my Powerbook.
I've spent the past five hours trying to get this mini to start and operate normally. From total shutdown to booted up and "ready" ( I use the term loosely) for use takes about 15 minutes. The first thing I noticed was the Logitech Bluetooth mouse was not working. I changed the batteries and now it seems to be working, although through every one of the approx. 30 reboots I've done today, the system initially cannot find the mouse, but manages to work after about four minutes of a spinning cursor. To launch an application, say, System Preferences, takes about five minutes from mouse click to mouse click.
I tried going to Disk Utility to try and repair the hard drive, but Repair Disk is grayed out. I tried using Software Update, figuring it would optimize the disk in the process. After downloading a list of available updates, I tried running only the Mac OS X update (OS X 10.4.8 now), but it broke with an error message saying the volume could not be updated. There is a file on the desktop named "file001.zip", which leads me to believe there is a corrupted file on the disk somewhere, but when I drag it to the trash I get an error message saying it cannot be deleted.
I pulled out the manual, and tried booting from the original install disk. I followed the instructions and tried repairing the disk from there, but I got an error message saying that wouldn't work either (memory getting kind of hazy; I've been at this quite a long time). In other words, nothing I try works. My deepest suspicion is that something is wrong with a volume, partition, partition table, partition map, or something like that. On the desktop is a disk icon named "Macintosh HD", and if I'm not mistaken it should have a user name instead, just like on my PB the same icon is named "Jims HD". I would be grateful for some other ideas or avenues to explore. I'm leaving town in the morning so if it's going to come down to erasing and reinstalling from the original disks, I'd better get started on that soon. Thanks.
Jim

Answer 1

The easy bit first - the 'Macintosh HD' label on the drive icon is perfectly OK. You can change the name to just about anything you want, but by default the drive is called Macintosh HD, and doesn't change to the user's name (or anything else) without the user's intervention.
As to the other issues - it does sound suspiciously like the drive has issues, or the MacOS install is malfunctioning. Unfortunately, you can't repair the drive you boot from, so while you can run Disk Utility from the hard drive and 'verify' it, to see if it reports any errors, you can only repair it by booting from the original install disk that came with the system (or another viable boot volume if you have one on an external drive for example).
It seems to me that unless the system has already accumulated user data and files, the easiest way to deal with this is simply to insert the original install disk and reboot with the C key down, then do an Erase and Install. Note that this will delete all data, so if there is anything on the system of importance, it will need to be backed up. After the erase and install, and before installing any software, run the system for a while to make sure it's OK, including Disk Utility to verify the hard drive is fully functional.
If you prefer to take the diagnostic approach, the first step is to boot to the original install disk and run Disk Utility from there, with the repair disk option selected for the internal drive. You will likely need a wired keyboard to get the system to boot from the optical drive because with many BT/cordless keyboards, the driver isn't loaded until AFTER the point at which the C key boot modifier stops being recognized.
If you can't get the system to boot to the install disk, or if Disk Utility reports errors it can't repair, post again with as much detail specific to the issue as you can.