The following code only changes the color around the drop down menu and not the entire cell.

Code:
Choose a background color for this table cell:<br />
       <select onChange="this.parentNode.style.backgroundColor = this.options[this.selectedIndex].value;" name="bg_color_cell2" size="1">
        <option value="">-- Select a color --</option>
        <option style="background-color: #ffffff;" value="#ffffff">#ffffff</option>
        <option style="background-color: #ffff99;" value="#ffff99">#ffff99</option>
        <option style="background-color: #ccff99;" value="#ccff99">#ccff99</option>
        <option style="background-color: #ffcc99;" value="#ffcc99">#ffcc99</option>
        <option style="background-color: #cccc33;" value="#cccc33">#cccc33</option>
        <option style="background-color: #66cc99;" value="#66cc99">#66cc99</option>
      </select>