Dynamic dropdowns won't help.
You probably want something like:
<select onchange="this.form.submit();" name="product" id="product" width="160" style="width: 160px">
<option value="" selected>Search by Product</option>
<?php
do {
?>
<option value="<?php echo $row_prods['ID']?>" ><?php echo $row_prods['Name']?></option>
<?php
} while ($row_prods = mysql_fetch_assoc($prods));
$rows = mysql_num_rows($prods);
if($rows > 0) {
mysql_data_seek($prods, 0);
$row_prods = mysql_fetch_assoc($prods);
}
?>
</select>