Thanks to a typically excellent tip from Adam Williamson (AdamW), I now have Mandriva One 2009.0 running on the HP 2133, and it looks really good. When I tried it previously, it appeared that the LiveCD didn't boot properly. It appears now that I might just not have been waiting long enough, because when I tried it again this morning, and paid careful attention to what it was doing so that I didn't give up too soon, the LiveCD did eventually come up, and the rest of the installation went very smoothly.
Mandriva is the only Linux distribution I have tried so far which recognizes the VIA Chrome9 graphic adapter, and set up X Windows to use the openchrome driver. Adam pointed out that the version distributed with Mandriva 2009.0 was still unstable, and gave me a link to a more stable version. I downloaded and installed that, and the display works absolutely flawlessly! For the first time, it picked up the 1280x768 resolution without a lot of struggling with the xorg.conf file, and it starts and runs smoothly and quickly, as evidenced by the messages in the Xorg.0.log file. It is truly wonderful - Adam, thank you very much!
I ran into one more problem, with the Broadcom 4312 wireless network adapter, but this time it was a lot easier to fix. First, because there in an article in the Mandriva 2009 Release Notes about it, and second because I had seen the same problem with openSuSE, and they even include a shell script to fix it. In a nutshell, the Broadcom wireless adapter requires a bit of firmware for the driver which the Linux distributions are not allowed to include. So what you have to do is go to the Linux Wireless web page and download a package from which you can then extract what you need. Or, if you have access to an openSuSE distribution, you can use the script /usr/sbin/install_bcm43xx_firmware to do the job for you. Once that is done, wireless works perfectly.
I will be continuing to use Mandriva on the HP, and see if everything else really is working as well as it appears at first glance. If it is, this may turn out to be my preferred Linux on this netbook. I'm also going to be looking into the updated openchrome display driver that I have installed - if I can get that installed and working on the other distributions, it would make life much easier.