<!DOCTYPE html>
<html>
<head>
<script src="https://code.jquery.com/jquery-3.5.0.min.js">
</script>
</head>
<body>
<p>
Select one of the proposed options:
<select id="selectId">
<option value="firstEl">First Element</option>
<option value="secondEl">Second Element</option>
</select>
</p>
<button onclick="addOption()">
Add Element
</button>
<script type="text/javascript">
function addOption() {
optText = 'New Element';
optValue = 'newElement';
$('#selectId').append(new Option(optText, optValue));
}
</script>
</body>
</html>