In Vue.js can a component detect when the slot content changes

To my knowledge, Vue does not provide a way to do this. However here are two approaches worth considering.

Watching the Slot's DOM for Changes

Use a MutationObserver to detect when the DOM in the <slot> changes. This requires no communication between components. Simply set up the observer during the mounted callback of your component.

Here's a snippet showing this approach in action:

Vue.component('container', {
  template: '#container',
  
  data: function() {
    return { number: 0, observer: null }
  },
  
  mounted: function() {
    // Create the observer (and what to do on changes...)
    this.observer = new MutationObserver(function(mutations) {
      this.number++;
    }.bind(this));
    
    // Setup the observer
    this.observer.observe(
      $(this.$el).find('.content')[0],
      { attributes: true, childList: true, characterData: true, subtree: true }
    );
  },
  
  beforeDestroy: function() {
    // Clean up
    this.observer.disconnect();
  }
});

var app = new Vue({
  el: '#app',

  data: { number: 0 },
  
  mounted: function() {
    //Update the element in the slot every second
    setInterval(function(){ this.number++; }.bind(this), 1000);
  }
});
.content, .container {
  margin: 5px;
  border: 1px solid black;
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/vue/2.5.13/vue.js"></script>

<template id="container">
  <div class="container">
    I am the container, and I have detected {{ number }} updates.
    <div class="content"><slot></slot></div>
  </div>
</template>

<div id="app">
  
  <container>
    I am the content, and I have been updated {{ number }} times.
  </container>
  
</div>

Using Emit

If a Vue component is responsible for changing the slot, then it is best to emit an event when that change occurs. This allows any other component to respond to the emitted event if needed.

To do this, use an empty Vue instance as a global event bus. Any component can emit/listen to events on the event bus. In your case, the parent component could emit an "updated-content" event, and the child component could react to it.

Here is a simple example:

// Use an empty Vue instance as an event bus
var bus = new Vue()

Vue.component('container', {
  template: '#container',

  data: function() {
    return { number: 0 }
  },

  methods: {
    increment: function() { this.number++; }
  },
  
  created: function() {
    // listen for the 'updated-content' event and react accordingly
    bus.$on('updated-content', this.increment);
  },
  
  beforeDestroy: function() {
    // Clean up
    bus.$off('updated-content', this.increment);
  }
});

var app = new Vue({
  el: '#app',

  data: { number: 0 },

  mounted: function() {
    //Update the element in the slot every second, 
    //  and emit an "updated-content" event
    setInterval(function(){ 
      this.number++;
      bus.$emit('updated-content');
    }.bind(this), 1000);
  }
});
.content, .container {
  margin: 5px;
  border: 1px solid black;
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/vue/2.5.13/vue.js"></script>

<template id="container">
  <div class="container">
    I am the container, and I have detected {{ number }} updates.
    <div class="content">
      <slot></slot>
    </div>
  </div>
</template>

<div id="app">

  <container>
    I am the content, and I have been updated {{ number }} times.
  </container>
  
</div>

As far as I understand Vue 2+, a component should be re-rendered when the slot content changes. In my case I had an error-message component that should hide until it has some slot content to show. At first I had this method attached to v-if on my component's root element (a computed property won't work, Vue doesn't appear to have reactivity on this.$slots).

checkForSlotContent() {
    let checkForContent = (hasContent, node) => {
        return hasContent || node.tag || (node.text && node.text.trim());
    }
    return this.$slots.default && this.$slots.default.reduce(checkForContent, false);
},

This works well whenever 99% of changes happen in the slot, including any addition or removal of DOM elements. The only edge case was usage like this:

<error-message> {{someErrorStringVariable}} </error-message>

Only a text node is being updated here, and for reasons still unclear to me, my method wouldn't fire. I fixed this case by hooking into beforeUpdate() and created(), leaving me with this for a full solution:

<script>
    export default {
        data() {
            return {
                hasSlotContent: false,
            }
        },
        methods: {
            checkForSlotContent() {
                let checkForContent = (hasContent, node) => {
                    return hasContent || node.tag || (node.text && node.text.trim());
                }
                return this.$slots.default && this.$slots.default.reduce(checkForContent, false);
            },
        },
        beforeUpdate() {
            this.hasSlotContent = this.checkForSlotContent();
        },
        created() {
            this.hasSlotContent = this.checkForSlotContent();
        }
    };
</script>