Switching from functional jQuery code to object-oriented
Your mistake is in your processInput() function, it is not created as part of the search object and is invalid syntax the way you have it written.

I try to avoid private functions in javascript as it makes them difficult to test, but if you really wanted to have private functions, you could write your module like this:

var AutoSearch = (function () {
    "use strict";

    // private
    var my = {},
        input;

    // private
    function processInput() {
        alert(input.val());
    }

    // private
    function bindEvents() {
        input.on('keyup', processInput());
    }

    // public
    my.init = function (options) {
        input = options.input;
        bindEvents();
    };

    return my;
}());


It's not obvious to me that you need jQuery or .proxy() at this point, since all the data and functions for the event handler are inside the closure.