The LyX error looked like this:
LyX: Creating directory
Same as entered returned
Failed to create directory. Exiting.
And Atlantis didn't give me any error message. So, after a long time fiddling around with directory permissions, changing my group from admin to user and chowning a bit, I noticed from a dtruss log of failed and successful (by simply running as another user with sudo) runs of the affected programs that the file ~/.CFUserTextEncoding was being accessed at some point.
So I had a quick look at this file in the failure (first) and success (...second) cases:
Macintosh:~ oisin$ more .CFUserTextEncoding
Macintosh:~ oisin$ more ../yunqing/.CFUserTextEncoding
../yunqing/.CFUserTextEncoding: Permission denied
Macintosh:~ oisin$ sudo more ../yunqing/.CFUserTextEncoding
Replacing mine with 0:0 and su'ing to myself to get a new shell, I was able to run LyX and Atlantis with no problems. So, I'm not sure what this means - maybe the crash wiped out an encoding method or wrote the wrong data to that file, and when applications read from .CFUserTextEncoding they tried to look for/save their settings in the wrong place.
Confusing, but at least there's a fix.