JSFiddle - React, Tailwind, and code Playground

HTML

<select id="mySelect">
    <option>value 1</option>
    <option value="[new]">new</option>
</select>

JavaScript

// get a cross-browser function for adding events, place this in [global] or somewhere you can access it
var on = (function(){
    if (window.addEventListener) {
        return function(target, type, listener){
            target.addEventListener(type, listener, false);
        };
    }
    else {
        return function(object, sEvent, fpNotify){
            object.attachEvent("on" + sEvent, fpNotify);
        };
    }
}());

var mySelect = document.getElementById("mySelect");    
on(mySelect, "change", function(){
    if (this.value === "[new]") {
        var newOptionName = prompt("input new option");
        if (newOptionName){
            mySelect.options[mySelect.options.length] = new Option(newOptionName);
            this.value= newOptionName;
        }
    }
});