Removing an item from a select box

Remove an option:

$("#selectBox option[value='option1']").remove();

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<select name="selectBox" id="selectBox">
  <option value="option1">option1</option>
  <option value="option2">option2</option>
  <option value="option3">option3</option>
  <option value="option4">option4</option>	
</select>

Add an option:

$("#selectBox").append('<option value="option5">option5</option>');

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<select name="selectBox" id="selectBox">
  <option value="option1">option1</option>
  <option value="option2">option2</option>
  <option value="option3">option3</option>
  <option value="option4">option4</option>	
</select>

You can delete the selected item with this:

$("#selectBox option:selected").remove();

This is useful if you have a list and not a dropdown.

window.onload = function ()
{   
    var select = document.getElementById('selectBox');
    var delButton = document.getElementById('delete');

    function remove()
    {
        value = select.selectedIndex;
        select.removeChild(select[value]);
    }

    delButton.onclick = remove;    
}

To add the item I would create second select box and:

var select2 = document.getElementById('selectBox2');
var addSelect = document.getElementById('addSelect');

function add()
{
    value1 = select2.selectedIndex;
    select.appendChild(select2[value1]);    
}

addSelect.onclick = add;

Not jQuery though.