Thanks, Freemor. I took your suggestion and tried adding an SD card for swapping. I got a good deal on a 4 GB SDHC card (<$10) and following the wiki instructions to set up for a 1GB swapping partition on it, turned swap on. Swapping did not prevent the "Abort" which happens immediately when I try to activate my XO outgoing video while receiving incoming video on Skype.
I read some postings elsewhere which said the problem may be due to lack of 2 xv ports for video overlay windows. The command xvinfo reports various parameters of the video chip. On the XO, it reports that there is only 1 xv port available. When the Skype program crashes, the error output shows that the first video reported 1 xv port available, but there is a second line farther down reporting zero (0) xv ports available (perhaps after the incoming video window used the first port). I looked at the specifications for the AMD Geode video processor that's in the XO and didn't find any mention of more than 1 video overlay port in the hardware. So it seems logical that the lack of a second xv port could cause Skype to abort when it tries to display both video windows at the same time.
However, that theory doesn't hold up. I tried Skype on a desktop computer running Ubuntu 8.04.1. It was the same version of Skype (126.96.36.199) for Linux as I downloaded on my XO. The xvinfo command on the desktop also reports only 1 xv port available. I'm running Skype with a Logitech QuickCam Pro 9000. The video works perfectly when outgoing and incoming video are displayed, rather than aborting as the XO does. I didn't have to do the gstfakevideo pipeline trick on the Ubuntu system, unlike the XO where that trick is necessary to make the built-in camera work with Skype, as described on the Skype page of wiki.laptop.org. The camera just works out of the box: plug it into the desktop computer, launch Skype, and 2-way video works.
This shows me that the problem is not in Skype nor in the lack of more than one xv port in the XO's video chip. It's a software problem in the XO's OS or in the gstfakevideo trick. I don't think it could be the slowness of the XO processor as you suggested, because in my opinion that would result in poor performance (jerky video, delays, etc.) not an error message from Skype saying it was aborting. So, I still can't do 2-way video on my XO with Skype.