Your variable names are very confusing. You're looking for $old_name in file $new?? In fact, I think you confused yourself. Use consistent, meaningful names.

Is this what you want?

sub fc {
   # Pass by reference for efficiency.
   local $_; *_ = \$_[1];
   return uc(substr($_, 0, 1)) . lc(substr($_, 1));
}


sub match_case {
   # Very simplistic.
   # Pass by reference for efficiency.
   our $template; *template = \$_[0];
   our $s;        *s        = \$_[1];

   return uc($s) if (uc($template) eq $template);

   my $first_char = substr($template, 0, 1);
   return fc($s) if (uc($first_char) eq $first_char);

   return lc($s);
}


sub wanted {
    if ((-f $File::Find::name) && ($File::Find::name !~ m/\.bak/)) {
        my $old_file_name = $File::Find::name;

        (my $new_file_name = $old_file_name) =~ s/$old_name/$mname/;

        # # Make backup
        # rename($new_file_name, $new_file_name.'.bak');

        local *IN;
        open(IN, '<' . $old_file_name)
        or die("Unable to open old file: $!$/");

        local *OUT;
        open(OUT, '>' . $new_file_name)
        or die("Unable to create new file: $!$/");

        while (<IN>) {
            s/($old_name)/match_case($1, $mname)/eig;
            print OUT $_;
        }

        close(OUT);
        close(IN);

        unlink($old_file_name);
    }
}
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Update: Added match_case.

Update: Added unlink as per later discussions.