BuddyPress.org

Some hosts allow the PHP memory limit to be changed via .htaccess, or via a WordPress configuration (see this). I don’t know about your specific host — I suggest contacting them and ask to increase the limit to 96Mb (as you appear to be already on 64Mb).

Before you do that, however, check in your web server’s error log for any warnings which are generated when you try to activate BuddyPress. Is there anything in there?