I haven't been doing Linux for long but i have messed about with a live dist, i can't see why you couldn't mount a USB mem stick and use that as a mini HDD.

/media/floppy or /cdrom why not /media/mem - save and run the sh boinc_5.2.13_i686-pc-linux-gnu.sh script in that location.

Not sure about live dists but if you want to create a mount point after a reboot edit the fstab file in /etc - be careful and make a backup.