Adding onclick attribute to a dynamically created option

A much better idea would be to put a click event handler on the select itself, then read the value which was selected.

This is for two reasons. Firstly inline event attributes, such as onclick, are no longer good practice and should be avoided where possible. Secondly, event handlers on option elements are unreliable at best and just plain don't work at worst.

The best way to achieve what you need is to add a change event handler to the select and read the selected value from the chosen option at that point, something like this:

let $foo = $('#foo').on('change', e => {
  console.log(e.target.value);
});

$('button').on('click', () => {
  let ts = (new Date()).getTime();
  $foo.append(`<option value="${ts}">${ts}</option>`);
});

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>

<button>Add option</button>
<select id="foo"></select>

I wouldn't do it for an onclick attribute since, as already stated, there are faster and better ways to reference it directly from JS. But for future reference or if you need to set other attributes, you can use the built in setAttribute. An example would be yourelement.setAttribute("class","classname").

After adding a new option, you could create a listener for that select like this:

jQuery('select').change(function(){ 
    const value = jQuery(this).val();
    console.log(value);
});

jQuery('select').append(new Option('New Option', 'value'));

jQuery('select').change(function(){ 
    const value = jQuery(this).val();
    console.log(value);
});

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<select>
  <option value="option1">Option 1</option>
  <option value="option2">Option 2</option>
</select>