This is a userscript (for Greasemonkey, Tampermonkey, etc.) for sorting answers to posts randomly, like so:

This feature was requested in connection with the Stack Overflow Fall 2015 Moderator Election Q&A - Questionnaire.

To install the script, just click here. You can also view the source (a github gist) and install by clicking the "Raw" button.

#Source:

// ==UserScript==
// @name         Stack Overflow Random Answer Sort Tab
// @namespace    https://github.com/edcottrell/
// @version      0.1.2
// @description  Add a "random" tab for sorting Stack Overflow answers
// @author       Ed Cottrell
// @license      MIT
// @match       *://*.askubuntu.com/*
// @match       *://*.mathoverflow.net/*
// @match       *://*.serverfault.com/*
// @match       *://*.stackapps.com/*
// @match       *://*.stackexchange.com/*
// @match       *://*.stackoverflow.com/*
// @match       *://*.superuser.com/*
// @exclude     *://data.stackexchange.com/*
// @exclude     *://stackexchange.com/users/*
// @grant        none
// ==/UserScript==

// set up some variables
var $answers = jQuery('div.answer'),
    arrayIndex,
    newOrder = [],
    prop,
    randomWeights = {},
    relativeUrl = window.location.href.replace(/^.+\/\/.+?\//, '\/'),
    selectedTab = window.location.href.replace(/.+?answertab=([^&#]+).*/, '$1'),
    tabUrl = relativeUrl.replace(/([?&]answertab=)[^&#]+/, '$1random')
    ;

// make sure the URL has an answertab parameter
if (!tabUrl.match(/([?&]answertab=)[^&#]+/)) {
    tabUrl = (tabUrl + '?answertab=random').replace(/(\?.+)\?/, '$1&');
}


// add a random sort tab
jQuery('#tabs a[data-value="oldest"]').after('<a href="' + tabUrl + '" title="Random" data-value="random" data-nav-xhref="">random</a>');

if (selectedTab === 'random') {
    // fix the tab highlighting
    jQuery('.youarehere').removeClass('youarehere');
    jQuery('a[title="Random"]').addClass('youarehere');

    // give each answer a random "weight" for sorting
    $answers.each(function (x) {
        randomWeights[$(this).prop('id').replace(/answer-/, '')] = Math.random();
    });

    // sort the ids by weight
    for (prop in randomWeights) {
        newOrder.push(prop);
    }
    newOrder.sort(
        function(a,b) {
            return randomWeights[a] > randomWeights[b] ? 1 : (randomWeights[a] == randomWeights[b] ? 0 : -1); 
        }
    );

    // remove the anchor tags and answers and append them in the right order
    for (arrayIndex = 0; arrayIndex < newOrder.length; arrayIndex++) {
        $('a[name=' + newOrder[arrayIndex] + ']')
            .detach()
            .appendTo('div#answers');
        $('div#answer-' + newOrder[arrayIndex])
            .detach()
            .appendTo('div#answers');
    }

    // reset locations of the non-answer elements of the #answers div
    $('#answers a[name="new-answer"]')
        .detach()
        .appendTo('div#answers');
    $('#post-form')
        .detach()
        .appendTo('div#answers');
    $('#show-editor-button')
        .detach()
        .appendTo('div#answers');
    $('#answers .bottom-notice')
        .detach()
        .appendTo('div#answers');
}

Caution: This is poorly (barely) tested and may melt your browser or make kittens cry. More likely, it will just slightly slow down SE-network sites, but I make no promises.

Any feedback would be appreciated!