Pretty much the idea, but you seem a little confused by variables. Lightly edited (changed to a print statement to protect my innocent file names), here is your code:

#!/usr/bin/perl
use strict;
use warnings;

dir('.');

sub dir {
    opendir( DIR, shift );
    my @list_of_files = readdir(DIR);
    foreach (@list_of_files) {
        if (-d $_) {
            dir($_) if $_ !~ /^\.\.?$/;
        }
        else {
            print $_," would become \L$_,\n";
        }
    }
}
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Things to note:

• If you mean to use a variable as a parameter, just do it, don't embed it in quotes
• Your use of $_[0] isn't needed, easier way to do the same is shown
• Post conditions and regexs are your friend, use them and they will reward you!

–hsm

"Never try to teach a pig to sing…it wastes your time and it annoys the pig."