How to efficiently randomly select array item without repeats?

When you instantiate Shuffler, give it your array as a parameter. It will create a copy of the array and every time next() is called it will return a random element from a copy and remove it from the copy array so that no repeats are possible.

var Shuffler = function(a) {
    var aCopy = [],
        n     = 0;

    // Clone array
    for (n=0; n<a.length; n++) {
        aCopy.push(a[n]);
    }

    this.next = function() {
        if (aCopy.length == 0) { return null; }

        var nRandom  = Math.floor(Math.random() * (aCopy.length + 1)),
            mElement = aCopy[nRandom];

        delete aCopy[nRandom];
        return mElement;
    }
}

var oShuffler   = new Shuffler([/* names go here */]),
    sRandomName = null;

while (sRandomName = oShuffler.next()) {
    console.log(sRandomName);
}

I recommend you to use underscore.js, it will be very simple.

The function shuffle is implemented in uniformly distributed way so the probability of repetition will be low if the array a contains more data.

var a = ["Roger", "Russell", "Clyde", "Egbert", "Clare", "Bobbie", "Simon", "Elizabeth", "Ted", "Caroline"];
b = _.shuffle(a).slice(0,5);
console.log(b);

Whenever an item is selected, move it to the back of the array and randomly select from a slice of the original array array.slice(0, -5).

var a = ["Roger", "Russell", "Clyde", "Egbert", "Clare", "Bobbie", "Simon", "Elizabeth", "Ted", "Caroline"];

var chooseName = function () {
    num = Math.floor(Math.random() * a.length - 5);
    name = a.splice(num,1);
    a.push(name);
}


window.addEventListener("keypress", function (e) {
    var keycode = e.keyCode;
    if (keycode == 13) {
        chooseName();
    }
}, false);

EDIT: This also has the side-effect of not giving whichever variables happen to tail the list the unfair disadvantage that they won't be considered in the first N calls. If that's a problem for you, maybe try hold a static variable somewhere to keep track of the size of the slice to use and max it out at B (in this case, 5). e.g.

var a = ["Roger", "Russell", "Clyde", "Egbert", "Clare", "Bobbie", "Simon", "Elizabeth", "Ted", "Caroline"];
B = 5; //max size of 'cache'
N = 0;

var chooseName = function () {
    num = Math.floor(Math.random() * a.length - N);
    N = Math.min(N + 1, B);
    name = a.splice(num,1);
    a.push(name);
}

I like commenter @YuriyGalanter's idea of choosing items randomly until all are taken and only then repeating, so here's an implementation:

function randomNoRepeats(array) {
  var copy = array.slice(0);
  return function() {
    if (copy.length < 1) { copy = array.slice(0); }
    var index = Math.floor(Math.random() * copy.length);
    var item = copy[index];
    copy.splice(index, 1);
    return item;
  };
}

var chooser = randomNoRepeats(['Foo', 'Bar', 'Gah']);
chooser(); // => "Bar"
chooser(); // => "Foo"
chooser(); // => "Gah"
chooser(); // => "Foo" -- only repeats once all items are exhausted.