JavaScript switch statement to make an AJAX call
I don't really think putting the URL in your HTML is a good idea; Separating presentation from functionality and all that...

However, instead of using a switch statement, you should be using an object literal to map the URLs to the clickedOptionId key:

var searchURLs = {
    'btUsername'    : 'SearchByUsername',
    'btLastName'    : 'SearchByLastName',
    'btStudentID'   : 'SearchByStudentID'
};

var search = {
    url: searchURLs[clickedOptionId],
    data: { criteria: $search.attr('value') }
};