I think my approach would be to generate the embedded text separately and interpolate, e.g. (untested):

my $text_to_interpolate = '';

for my $i (0 .. $#system_names) {

    my ($color_0, $color_1) = ('#a1a1a1', '#f3f3f3');
    $rowcolor = $i % 2 ? $color_1 : $color_0;

    $text_to_interpolate .= qq{    
        <tr bgcolor="$rowcolor">
          <td width=""> $system_names[$i] </td>
          <td width=""> $name_names  [$i] </td>
          <td width=""> $menu_names  [$i] </td>
          <td width=""> $option_names[$i] </td>
          <td width=""> $system_names[$i] </td>
          <td width=""> Received          </td>
        </tr>
        };      

    }

...

  Data => qq|
    <table width ...> $text_to_interpolate ...
    |,
...
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If it is necessary to interpolate directly into a double-quoted string, use this:

my $string = "before @{[ code() . "stuff" ]} after";
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Note the "string @{[ code ]} string" idiom.