There are multiple bugs in there. Off the top of my head, the worst ones are:

1. You're not assigning the return from "generatePassword" to a variable, so there's no way that "pleaseGodWork" can do anything with it.
2. You need to make the change to the appropriate line in @copy_this, e.g.

     for (@copy_this) {
         s{ \*\*\* password \*\*\* }{$password}xms;
     }
   [download]

3. the "open (MYFILE" line is missing the '<' you need to indicate read access, and while you're at it you might as well be using open in the modern style:

         open my $myfile_fh, '<', $readfile or die "Can't open $readfile 
   +for input: $!"
   [download]

Further, it makes no sense to do those "close" operations after the end of the sub. If you're going to do them explicitly at all, put them inside the sub.

There are a number of other stylistic objections that could be made: I would name the subs "generate_password" and "please_god_work" myself, and I'd probably slurp the whole file into one string like so: undef $/; my $myfile = <$myfile_fh>;, so then the change could be made in one step  $myfile =~ s{\*\*\* password \*\*\*}{$password}xms;

Perhaps worst of all, you appear to be in the process of inventing your own system of html templates, and you'd really be better off using an existing one, e.g. Mason or Template::Toolkit.

the "open (MYFILE" line is missing the '<' you need to indicate read access

This may be considered poor style, but it's not exactly a bug. '<' is optional.

You might like to consider using a a templating system such as HTML::Template. Consider:

use strict;
use warnings;
use HTML::Template;

# Create a sample file - for demo purposes only
open OUTFILE, '>', 'sample.tmpl';
print OUTFILE <<HTML;
<p>The password is: <TMPL_VAR NAME='password'></p>
HTML
close OUTFILE;

# Here be the important sample code
my $template = HTML::Template->new (filename => 'sample.tmpl');

$template->param (password => 'wibble');
print $template->output ();
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Prints:

<p>The password is: wibble</p>
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I don't understand why you close the file handles MYFILE and YOURFILE before you actually open them in pleaseGodWork() subroutine. It's good to close the handles in the same subroutine as you open them.

Now you might want to use something like HTML::Parser module in-order to fix the password field in the index.txt file to replace it(In the current version there is no replacement ).

sub random_string {
    local $" = '';
    "@{[map{chr(97+int rand 26)} 1 .. shift]}";
}
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local $" = ''; "@{[ ... ]}"
instead of
join '', ...?

sub random_string {
    join '', map{chr(97+int rand 26)} 1 .. shift;
}
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{   my @possible = ( 'a' .. 'k', 'm', 'n', 'p' .. 'z', 2 .. 9, 'A' .. 
+'H', 'J' .. 'N', 'P' .. 'Z' );

    sub generatePassword {
        my $length = shift;
        return join '', map $possible[ rand @possible ], 1 .. $length;
        }
    }


{   my $readfile  = 'index.txt';
    my $writefile = 'index.html';

    sub pleaseGodWork {
        my $password = generatePassword 8;
 
        open my $IN, '<', $readfile or die "Cannot open $readfile: $!"
+;
        open my $OUT, '>', $writefile or die "Cannot open $writefile: 
+$!";
        select $OUT;

        s/\Q***password***/$password/g, print while <$IN>;
        }
    }

pleaseGodWork;
exit 0;
[download]

So the generated password is recorded in a file as it is. What kind of shit is this? So anyone take a look at the file will know the password.

If this is a real system, nobody, including you or the system admin is supposed to know this.

Thanks everyone for the help. I was half-expecting to be brutally made fun of and laughed at for the sloppy coding. ;-)
I will try some of these sugesstions.
Cop: The randomly generated password is going to change every day. The file gets overwritten once a day with a new password. This password will reside in a password-protected web area for the admins of that particular to view and assign to users.