OK, this seems to work, setting --exec to the name of Perl, and --startas to the name of the script (one last try did it, apparently):

#! /bin/sh

PATH=/usr/local/sbin:/usr/local/bin:/sbin:/bin:/usr/sbin:/usr/bin
DAEMON=/home/wander/wander/LCD/monitor.pl
NAME=wanderLCD
DESC="Wander LCD driver"

test -f $DAEMON || exit 0

set -e

case "$1" in
  start)
 echo -n "Starting $DESC: "
 start-stop-daemon --start --quiet --pidfile /var/run/$NAME.pid \
  --exec /usr/bin/perl --startas $DAEMON
 echo "$NAME."
 ;;
  stop)
 echo -n "Stopping $DESC: "
 # --quiet
 start-stop-daemon --stop --signal 15 --pidfile /var/run/$NAME.pid \
  --exec /usr/bin/perl --startas $DAEMON
 echo "$NAME."
 ;;
  restart|force-reload)
 echo -n "Restarting $DESC: "
 start-stop-daemon --stop --quiet --pidfile \
  /var/run/$NAME.pid --exec /usr/bin/perl --startas $DAEMON
 sleep 1
 start-stop-daemon --start --quiet --pidfile \
  /var/run/$NAME.pid --exec /usr/bin/perl --startas $DAEMON
 echo "$NAME."
 ;;
  *)
 N=/etc/init.d/$NAME
 # echo "Usage: $N {start|stop|restart|reload|force-reload}" >&2
 echo "Usage: $N {start|stop|restart|force-reload}" >&2
 exit 1
 ;;
esac

exit 0
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In the Perl script, I have something like:

BEGIN {
 # Fork.
 my $pidFile = '/var/run/wanderLCD.pid';
 my $pid = fork;
 if ($pid) # parent: save PID
 {
  open PIDFILE, ">$pidFile" or die "can't open $pidFile: $!\n";
  print PIDFILE $pid;
  close PIDFILE;
  exit 0;
 }
}
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