On the tonido support forums (requires login, but I'm posting the link here anyway) aleinss helpfully pointed at Logging into tonido as a non root user.
Apparently, with 2.6.31 kernels and later, /proc/sys/vm/mmap_min_addr needs to be 32768 (instead of the previous 65536).
I tested it with sudo echo "32768" > /proc/sys/vm/mmap_min_addr but that didn't work. Reboot required, I guess. The solution was to edit /etc/sysctl.d/10-process-security.conf and edit the vm.mmap_min_addr line to say
vm.mmap_min_addr = 32768
Many thanks to aleinss for pointing that out.