I don't think that lighttpd or logrotate can do this by themselves. I see two options to achieve this:
Do your own scripting directly after logrotate, like at the end of it's cronjob. Something like:
if [ -f /var/log/lighttpd/access.log.1.gz ]; then`
mv /var/log/lighttpd/access.log.1.gz /var/log/lighttpd/access.log.$date.gz
fi
Or, syslog-ng can build log files by date, and lighttpd can send logs to syslog rather than writing them itself.
# syslog-ng.conf
destination df_lighttpd { file("/var/log/lighttpd/$YEAR$MONTH$DAY.log"); };
filter f_lighttpd { program("lighttpd"); };
log { source(s_all); filter(f_lighttpd); destination(df_lighttpd); };
and set accesslog.use-syslog
& server.errorlog-use-syslog
in the lighttpd conf.
Edit: Cronolog is simpler and/or than either of my suggestions, I'd go with that if it works for you.