How can I have clickable text in the AppBar in Flutter

Use TextButton:

appBar: AppBar(
  actions: <Widget>[
    TextButton(
      onPressed: () {},
      child: Text('Save'),
    ),
  ],
)

You can use FlatButton within an AppBar's actions list:

appBar: AppBar(
  title: Text("Test Screen"),
  actions: <Widget>[
    FlatButton(
      textColor: Colors.white,
      onPressed: () {},
      child: Text("Save"),
      shape: CircleBorder(side: BorderSide(color: Colors.transparent)),
    ),
  ],
),

onPressed must be defined or the button will appear disabled. Also note that by default the shape of the button will create a filled rectangle for the InkWell effect. By setting the shape property to CircleBorder, we get a nicer effect for the pressed state.

E.g.

Not pressed:

Pressed: