I had to do the same thing on one of my sites.
Firstly, you should create a Template Override for the mod_login module
Once done, you should have the following file:
ROOT/templates/YOUR_TEMPLATE/html/mod_login/default_logout.php
Note: The path may be slightly different depending on the template's framework.
Then, Open the file mentioned above and add the following code
$doc = JFactory::getDocument();
$doc->addScriptDeclaration('
jQuery(document).ready(function($){
$("#nav").find(".item-222 a").empty().text("Logout");
});
');
You will of course need to change #nav
to your menu wrapped selector and .item-222 a
to the menu item selector.
If you provide a link to your site, I can give you the exact code used above, but until then, this is all I can provide.
Hope this helps