Random Colorful letters

If you want to get only vivid colors, using randomness like this wont do.

What you need is a function for HSV to RGB conversion, like this:

function HSVtoRGB(h, s, v)
{
    var r, g, b, i, f, p, q, t;
    i = Math.floor(h * 6);
    f = h * 6 - i;
    p = v * (1 - s);
    q = v * (1 - f * s);
    t = v * (1 - (1 - f) * s);
    switch (i % 6)
    {
        case 0: r = v, g = t, b = p; break;
        case 1: r = q, g = v, b = p; break;
        case 2: r = p, g = v, b = t; break;
        case 3: r = p, g = q, b = v; break;
        case 4: r = t, g = p, b = v; break;
        case 5: r = v, g = p, b = q; break;
    }
    r = Math.round(r * 255).toString(16);
    g = Math.round(g * 255).toString(16);
    b = Math.round(b * 255).toString(16);
    if (r.length < 2) r = '0' + r;
    if (g.length < 2) g = '0' + g;
    if (b.length < 2) b = '0' + b;

    return '#' + r + g + b;
}

Parameters range from 0 to 1. Using HSVtoRGB(Math.random(), 1, 1) will always yield you a vivid color with maximum value and saturation.

You would have to break your text into children spans and do the coloring on them.

function colorElement(element) {
  var randomColor = Math.floor(Math.random()*16777215).toString(16);
  element.style.color = '#' + randomColor;
}

function splitElement(element) {
  var text = element.innerText;
  element.innerText = '';
  var chars = text.split('');
  for(var ch of chars) {
     var charSpan = document.createElement('span');
     charSpan.innerText = ch;
     element.appendChild(charSpan);
  }
}

function randomColor() {
   var result = document.querySelectorAll('span span');
   for(ele of result){
      colorElement(ele);
   }
}

var ele = document.getElementById('myText');
splitElement(ele);
setInterval(function() {
   randomColor();
},500);

<div>
  <span id="myText">Disco Text</span>
</div>