ERRO: "The method 'setState' isn't defined for the type 'HomePage'."

Boa noite, seu HomePage é um statelessWidget, este não contem estado. Altere a classe para um statefulwidget que vai dar certo.

Marcos, obrigado pela resposta, mas mesmo assim o erro persiste. Por gentileza, veja abaixo:

import 'package:flutter/material.dart';

void main() {
  runApp(MyApp());
}

var theme = darkTheme;

var darkTheme = ThemeData(
  primarySwatch: Colors.red,
  brightness: Brightness.dark,
  visualDensity: VisualDensity.adaptivePlatformDensity,
);

var lightTheme = ThemeData(
  primarySwatch: Colors.teal,
  brightness: Brightness.light,
  visualDensity: VisualDensity.adaptivePlatformDensity,
);

class MyApp extends StatefulWidget {
  @override
  _MyAppState createState() => _MyAppState();
}

class _MyAppState extends State<MyApp> with WidgetsBindingObserver {
  int currentIndex =0;
  @override
  void initState() {
    super.initState();
    WidgetsBinding.instance.addObserver(this);
    changeTheme();
  }

  @override
  void dispose() {
    WidgetsBinding.instance.removeObserver(this);
    super.dispose();
  }

  @override
  void didChangePlatformBrightness() {
    changeTheme();
  }

  changeTheme() {
    var brightness = WidgetsBinding.instance.window.platformBrightness;
    brightness == Brightness.dark ? theme = darkTheme : theme = lightTheme;

    setState(() {});
  }

  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: 'Flutter Themes',
      debugShowCheckedModeBanner: false,
      theme: theme,
      home: HomePage(),
    );
  }
}

class HomePage extends StatefulWidget {
  int currentIndex = 0;
  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        title: Text("Home"),
      ),
      body: Center(
        child: Text("Hello"),
      ),
      bottomNavigationBar: BottomNavigationBar (
        currentIndex: currentIndex,
        onTap: (index) => setState(() => currentIndex = index),
        items: [
          BottomNavigationBarItem(
            icon: Icon(Icons.home),
            label: 'Home',
            backgroundColor: Colors.blue,
          ),
          BottomNavigationBarItem(
            icon: Icon(Icons.favorite),
            label: 'Favoritos',
            backgroundColor: Colors.red,
          ),
          BottomNavigationBarItem(
            icon: Icon(Icons.search),
            label: 'Pesquisar',
            backgroundColor: Colors.purple,
          ),
          BottomNavigationBarItem(
            icon: Icon(Icons.menu),
            label: 'Opções',
            backgroundColor: Colors.green,
          ),
        ],
      ),
    );
  }

  @override
  State<StatefulWidget> createState() {
    // TODO: implement createState
    throw UnimplementedError();
  }
}

Bom dia, desculpe a demora para responder, eu havia ido dormir kkkkkk.

A classe HomePage que deve controlar a alteração do do estado (que usa setState) precisa ser um statefulwidget. Somente o isto não o torna possível alterar o estado, para isto você precisa criar uma segunda classe que irá estender de State, nela que você coloca seu método build que retorna o widget e o seu estado, no caso o index do bottomNavigationBar. Com isto você pode chamar o setState (que é um método da classe State).

import 'package:flutter/material.dart';

void main() {
  runApp(MyApp());
}

var theme = darkTheme;

var darkTheme = ThemeData(
  primarySwatch: Colors.red,
  brightness: Brightness.dark,
  visualDensity: VisualDensity.adaptivePlatformDensity,
);

var lightTheme = ThemeData(
  primarySwatch: Colors.teal,
  brightness: Brightness.light,
  visualDensity: VisualDensity.adaptivePlatformDensity,
);

class MyApp extends StatefulWidget {
  @override
  createState() => _MyAppState();
}

class _MyAppState extends State<MyApp> with WidgetsBindingObserver {
  int currentIndex = 0;
  @override
  void initState() {
    super.initState();
    WidgetsBinding.instance.addObserver(this);
    changeTheme();
  }

  @override
  void dispose() {
    WidgetsBinding.instance.removeObserver(this);
    super.dispose();
  }

  @override
  void didChangePlatformBrightness() {
    changeTheme();
  }

  changeTheme() {
    var brightness = WidgetsBinding.instance.window.platformBrightness;
    brightness == Brightness.dark ? theme = darkTheme : theme = lightTheme;

    setState(() {});
  }

  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: 'Flutter Themes',
      debugShowCheckedModeBanner: false,
      theme: theme,
      home: const HomePage(),
    );
  }
}

class HomePage extends StatefulWidget {
  const HomePage({Key? key}) : super(key: key);

  @override
  createState() => _HomePageState();
}

class _HomePageState extends State<HomePage> {
  int currentIndex = 0;
  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        title: const Text("Home"),
      ),
      body: const Center(
        child: Text("Hello"),
      ),
      bottomNavigationBar: BottomNavigationBar(
        currentIndex: currentIndex,
        onTap: (index) => setState(() => currentIndex = index),
        items: const [
          BottomNavigationBarItem(
            icon: Icon(Icons.home),
            label: 'Home',
            backgroundColor: Colors.blue,
          ),
          BottomNavigationBarItem(
            icon: Icon(Icons.favorite),
            label: 'Favoritos',
            backgroundColor: Colors.red,
          ),
          BottomNavigationBarItem(
            icon: Icon(Icons.search),
            label: 'Pesquisar',
            backgroundColor: Colors.purple,
          ),
          BottomNavigationBarItem(
            icon: Icon(Icons.menu),
            label: 'Opções',
            backgroundColor: Colors.green,
          ),
        ],
      ),
    );
  }
}

Se você apertar ALT+ENTER com o stateless widget selecionado, ele te da a opção de transofrmar para stateful direto!

Muito obrigado, resolveu o meu problema.