How can I add a border to a widget in Flutter?

You can add the Text as a child to a Container that has a BoxDecoration with border property:

Container(
  margin: const EdgeInsets.all(15.0),
  padding: const EdgeInsets.all(3.0),
  decoration: BoxDecoration(
    border: Border.all(color: Colors.blueAccent)
  ),
  child: Text('My Awesome Border'),
)