Thanks for trying to help, folks, but even the 4-fold PRAM reset didn't do the trick. I followed that "Admin Hack" link and tried it that way, 'cause the stubborn thing just kept bringing me back to that logon screen, but even after changing the amount of RAM and doing the cmd-option-p-r thing 4 times, it still wouldn't come up in single-user mode.
I got totally sick of looking at that logon screen, and in the end, I was pretty sure I had recalled the right password, and it still wouldn't accept it. So, this afternoon I took it to a place in San Jose called Mac Pro, where they say they can wipe the hard drive and re-install OS 10.5x clean for just $25. All I wanted to do was give this old Mac to a friend, so I think $25 is a fair price.
If y'all are interested, I'll get back with you on Monday when I pick up the machine with whatever they say was wrong with it. Two possible complications are:
1. It sat in a closet for over a year, and when I fired it up a few months ago, the battery had gone bad and it wouldn't keep time or settings. When I replaced the battery, nothing changed, so I took it back to the store and they said the replacement battery was dead, too. The third battery they put on a tester, so we verified that it was good before I brought it home. Still, I was seeing that warning message whenever I'd start up the Mac, although it was apparently resetting the clock with the internet, as I had it on my Airport network. So, I suspect that something may have been shorting out the batteries . . . or,
2. I had wiped the personal data off my hard drive. I had done this by adding a second, smaller-capacity hard drive that I had in a drawer, wiping it using disk utility, and then copying the system onto it and using it as my startup drive, while I erased the larger one. At least, that's what I think I did. It sat in the corner and didn't get used again for another 2 months (my friend never got back to me) until a neighbor expressed an interest. That's when I turned it back on and couldn't get in. So, perhaps I deleted some essential file having to do with the single-user settings . . . anyway,
I took it in, and before I left I warned the technicians about the suspected battery problem, but they seemed to not be concerned. We'll see what they say on Monday. Again, thanks for your responses!