Make next post button link to random post

The code you posted does nothing but display. The code you need to alter is whatever is creating $prevID and $nextID, but that might not be entirely relevant.

Basically, you just need to grab a random post.

$rl = new WP_Query(
  array(
    'posts_per_page' => 1,
    'orderby' => 'rand',
    'fields' => 'ids'
  )
); ?>
<div class="alignleft">
  <a href="https://wordpress.stackexchange.com/questions/129453/<?php echo get_permalink($rl->posts[0]); ?>" title="<?php echo get_the_title($rl->posts[0]); ?>">Random</a>
</div><?php

I don’t see the point of two random post links but the same technique will work.

Also, using random links in your navigation isn’t really navigation. It doesn’t help your users navigate. It just randomly tosses them around the site.