Yeah, I know this sounds dumb, but the awstats perl program doesn't work through cgi in Lighttpd 1.5 (tested with revision 1878). Works fine under Lighttpd 1.4.
I wrote a short ruby cgi script and that works fine. I can see that perl is being executed, and that awstats is actually running (i put some debug code in there to test) but I just get an empty response. I get normal header, 200 code, empty body.
A couple of other people have had the same problems:
http://forum.lighttpd.net/topic/8283
I posted some debug output from my server there too.
I've tried disabling chunked encoding, and deflate compression. I've tested with and without ssl too.