Thank you for your response. It would have been very helpful if I hadn't kept working at it last evening!
In fact I had already tried zapping the PRAM before I posted on the board and that did not work.
However, soon after I posted, I was able to get booted in safe mode and made two separate backups.
diskutil did not find any problems with the startup disk, which could possibly be due to the fact that safe mode does a directory scan as part of the startup process.
I was then able to get it to do a full reboot, but VERY slowly, so something is still not working properly, even though I can now get past the blue screen.
There was another thread that suggested that this is a video card problem and that it has been successfully solved by deleting the extensions related to the video card, but I haven't tried that as yet.
Oh, and as for the OS version - since it is not actually my own computer, and since at the time I posted I wasn't able to get it booted, I had no way to know what OS X version it was running. I thought it was 10.6 but in fact when I did get it booted, it turned out to be 10.5.8
The last thing that you mention is to "insert the install disk" which would normally have been my first step, but this is a Macbook Air... No disk drive. And no way to boot in target disk mode to access the data.