vue.js - change text within a button after an event

Here you have your code in a snipplet.

I change your button by a plain object instead of an array and small adaptation in method toggleSeen.

// components
Vue.component('todoitem', {
  template: "<li>Test Item</li>"
})

// app code
var app = new Vue({
  el: '#app',
  data: {
    todos: [
      { text: 'Sample Item 1' },
      { text: 'Sample Item 2' },
      { text: 'Sample Item 3' }
    ],
    button: {
      text: 'Hide'
    },
    seen: true
  },
  methods: {
    addItem: function() {
      let item = document.getElementById("list-input").value;
      let error = document.getElementById("error");
      if (item == "") {
        error.style.display = "block";
      } else {
        app.todos.push({ text: item });
        error.style.display = "none";
      }
    },
    toggleSeen: function() {
      app.seen = !app.seen;
      app.button.text = app.seen ? 'Hide' : 'Show';
    }
  }
});
<script src="https://vuejs.org/js/vue.min.js"></script>

<div id="app">
  <ul>
    <li v-for="todo in todos">
      {{ todo.text }}
    </li>
  </ul>

  <input type="text" id="list-input">
  <input type="submit" id="list-submit" v-on:click="addItem">
  <span id="error" style="color: red; display: none;">Please Enter Text</span>

  <ul>
    <todoitem></todoitem>
    <todoitem></todoitem>
    <todoitem></todoitem>
  </ul>

  <h2 v-if="seen">SEEN</h2>
  <button id="hide-seen" v-on:click="toggleSeen">{{ button.text }}</button>
</div>

You can achieve this by using refs in vuejs:

<body>
    <div id = 'app'>
        <button @click="changeState" ref="btnToggle">Hide</button>
        <div v-show="show">
            <h1>1 to 100</h1>
            <p v-for="i in 100">{{i}}</p>
        </div>
    </div>
    <script>
        const app = new Vue({
            el:'#app',
            data: function(){
                return{
                    show: true
                }
            },
            methods: {
                changeState: function(){
                    this.show = !this.show;
                    this.$refs.btnToggle.innerText = this.show?'Hide':'Show';

                }
            },
        });
    </script>
</body>