Fairly good, few questions/problems
1) Why are you making the a "master" file if your goal is to randomize for mapcycle.txt
2) Put your file open/close statements outside of the loop.
3) Strict and warnings, but that's just me

Here's what I'd do:(comments removed)

#!/usr/bin/perl -w
#i#############################################
# Dave K 1/17/02                              #
#                                             #
# Use this script to randomize a HL map cycle #
###############################################

use strict;

my (@maps, @files, @reordered, $bsp);
my $mapsdir = "/Path/to/HL/maps/directory/";
my $cycle = "/Path/to/mapcycle.txt";

opendir(MAPSDIR, $mapsdir) || die "Can not open $mapsdir: $!\n";
@files = readdir(MAPSDIR);
closedir(MAPSDIR);

foreach $bsp (@files) {
        unless ( ($bsp eq ".") || ($bsp eq "..") ) {
                push @maps, $bsp;
        }
}

@reordered = shuffle(@maps);

open (MAPCYCLE, ">$cycle") || die "Can not open file $cycle: $!\n";
foreach (@reordered) {
        print MAPCYCLE  $_;
}
close (MAPCYCLE);

sub shuffle {
        my $array = @_;
        my $i = scalar (@$array);
        my $j;
        my $item;
        foreach $item (@$array)
        {
                --$i;
                $j = int rand ($i+1);
                next if $i == $j;
                @$array [$i,$j] = @$array[$j,$i];
        }
        return @$array;
}
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