I don't know that there is any simple way to do this with css, but you can do this fairly easily with jquery.
<a href="#" class="random-color">Test link 1</a>
<a href="#" class="random-color">Test link 2</a>
<!-- source script for jquery -->
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.4.1/jquery.min.js"></script>
<script>
  // Colors array with whatever colors you want
  var colors = ['#ff8800','#ff00ff','#00ffcc'];
  // This applies a random color to the link when you hover it
  $('.random-color').mouseenter(function(){
    $(this).css('color', colors[ Math.floor( Math.random() * colors.length ) ]);
  });
  // This function will return the color of the link to default when the mouse leaves
  $('.random-color').mouseleave(function(){
    $(this).css('color', '#0000ee');
  });
</script>
You can see the working code here.
The links are assigned a class, in this case random-color, which is used to access it via the jquery selector $('.random-color'). the .mouseenter event triggers when your mouse is over the link, and then using $(this).css() you can apply the styles you need. 
Using the colors[ Math.floor( Math.random() * colors.length ) ], you assign it a random element from the colors array. You can also add, subtract, or alter any of the colors in the array.