How to Set Dropdown Value with dom-repeat

I am trying to select the second option form a dropdown if specific conditions are met, am having problems with the selectedIndex

<select id="contact" on-change="selectContact">
  <option  value="-1" selected>Select Contact</option>
  <template is="dom-repeat" items="[[contacts]]" as="contact" index-as="index">
     <option value="[[contact]]">[[contact]]</option>
  </template>
</select>

<select id="name" on-change="selectName">
  <option  value="-1" selected>Select Name</option>
    <template is="dom-repeat" items="[[names]]" as="name">
      <option value="[[name]]">[[name]]</option>
    </template>
</select>

...

selectContact() {
  for (let i = 0; i < this.customerTable[0].length; i++) {
    if(true) {
        array[i] = this.customerTable[0][i]['name'];
      }
  }
  this.names = this.$.util.utilFn(array);


  if(this.names.length == 1) {
    this.shadowRoot.querySelector('#name').selectedIndex = 2;
  }

}

How can I select the second child of a dom-repeat dropdown?

In selectContact(), you're setting this.names (to which the second <dom-repeat>.items is bound) and immediately attempting to select the first item of that <dom-repeat> before the DOM has actually updated.

You actually need to wait until the next render frame with Polymer.RenderStatus.afterNextRender() before selecting the item:

selectContact() {
  this.names = ['John'];

  if (this.names.length === 1) {
    Polymer.RenderStatus.afterNextRender(this, () => {
      this.shadowRoot.querySelector('#name').selectedIndex = 1;
    });
  }
}

demo