How to store username after logout and show it in TextInput on next login?

you can use AsyncStorage component of React Native. For more info AsyncStorage

You can set a value of user on logout.

AsyncStorage.setItem('UseName', 'name of user');

On login form get value from AsyncStorage in componentWillMount() method.

 AsyncStorage.getItem('UserName')
      .then((data) => {
            if (data) {
                this.setState({UserName:data}) //store data in state
                                }
       }).done();