To do the simple dropdown search using JQquery, please follow below code, In this sample, I have used select2 jQuery plugin to create dropdown with a search box.

Please include select2.min.css and select2.min.js with jQuery library in the head section as follows:
<script src=""></script>
<link href="" rel="stylesheet" />
<script src=""></script>

Create a dropdown element and add few sample options and also add a button to read the selected dropdown value and dispay in the result div.
<!-- Sample Dropdown -->       
<select id="selectColor" style="width: 200px;" tabindex="-1" class="select2-hidden-accessible" aria-hidden="true">
   <option value="0">-- Select Color --</option>
   <option value="#9400D3">Violet</option>
   <option value="#4B0082">Indigo</option>
   <option value="#0000FF">Blue</option>
   <option value="#00FF00">Green</option>
   <option value="#FFFF00">Yellow</option>
   <option value="#FF7F00">Orange</option>
   <option value="#FF0000">Red</option>
<input type="button" value="Seleted Color" id="btnSelectColor">
<div id="result">Hexa Code : , Color  Name :  </div>
Please find below the JQuery script
<!-- Script -->
<script >
	$(document).ready(function () {

		// Initialize select2

		// Read selected option
		$('#btnSelectColor').click(function () {
			var colorName = $('#selectColor option:selected').text();
			var colorHexaCode = $('#selectColor').val();
			$('#result').html("Hexa Code : " + colorHexaCode + ", Color  Name : " + colorName);
	}); <

Please find the demo below and click here to download the sample html file.