(Before making the following changes, make sure to have a current backup of your files and your database.) 

You're going to do three steps: A) edit the configure.php settings and upload them, B) rename the admin folder, C) test login to the new folder. 
Details are below:

DO NOT USE SEARCH-AND-REPLACE TO DO THESE EDITS!!!!!!!!!!!

A - Edit /admin/includes/configure.php

Using your FTP program, download a copy of your /admin/includes/configure.php file to your computer.
Using a simple text editor like notepad (or better yet, use Notepad++ or TextWrangler), change all instances of admin to your chosen new admin folder-name.

For maximum security, you may want to consider that new folder name should include numbers and a combination of upper and lower case letters. The longer you make this folder's name the more secure it will be.

When editing, make sure you leave all the / (slashes) alone.

Change this section:

 define('DIR_WS_ADMIN', '/admin/');
 define('DIR_WS_CATALOG', '/');
 define('DIR_WS_HTTPS_ADMIN', '/admin/');
 define('DIR_WS_HTTPS_CATALOG', '/');

And this section:

 define('DIR_FS_ADMIN', '/home/mystore.com/www/public/admin/');
 define('DIR_FS_CATALOG', '/home/mystore.com/www/public/');

You will end up with something that looks like this:

 define('DIR_WS_ADMIN', '/mysecretadminarea/');
 define('DIR_WS_CATALOG', '/');
 define('DIR_WS_HTTPS_ADMIN', '/mysecretadminarea/');
 define('DIR_WS_HTTPS_CATALOG', '/');

And:

 define('DIR_FS_ADMIN', '/home/mystore.com/www/public/mysecretadminarea/');
 define('DIR_FS_CATALOG', '/home/mystore.com/www/public/');

B - Rename the Admin folder

Using your FTP software or your webhost's File Manager, find your Zen Cart™ /admin/ directory. Rename the directory to match the settings you just made in step A.

C - Login to your admin using the new URL

To login to your admin system you will now have to visit a new URL that matches the new name used in steps A and B above.

For example instead of visiting http://www.example.com/admin/ visit http://www.example.com/NeW_NamE4u/

Use of SSL is highly recommended to protect your and your customers information.