To make it work you need to add new edit handler, similar to pressHandler, but editing an entry instead of removing it. The possible edit handler could look like:
const editHandler = (todoKey, newText) => {
  const newTodos = [...todos];
  const index = newTodos.findIndex(todos => todos.key === todoKey);
  newTodos[index] = Object.assign(newTodos[index], { value: newText });
  setTodos(newTodos);
};
It moves the edited element at the end of the list. If you want, you can change this behavior on your own.
Then you need to pass the handler to <TodoItem />:
<TodoItem
  key={item.key}
  todoKey={item.key}
  title={item.value}
  editHandler={editHandler}
  pressHandler={pressHandler}
/>
You don't need to bind function component functions, but you need to provide a key props for every component you render in map(). I've changed it and provided a todoKey props I later use in <TodoItem />.
In <TodoItem /> you can similar logic for modifing todo text as you use in <AddTodo /> for creating new todo. I use conditional rendering to render the <TextInput /> when isEditing is true, and <Text /> when it's not.
{isEditing 
  ? <TextInput value={text} onChangeText={setText} style={styles.itemText} />
  : <Text style={styles.itemText}>{props.title}</Text>
}
Similarly, I conditionally render Save and Edit buttons.
Full <TodoItem /> component:
const TodoItem = props => {
  const [text, setText] = useState("");
  const [isEditing, setEdit] = useState(false);
  const handleEdit = () => {
    props.editHandler(props.todoKey, text);
    setText("");
    setEdit(false);
  };
  return (
    <View style={styles.items}>
      <View style={styles.itemContainer}>
        {isEditing 
          ? <TextInput value={text} onChangeText={setText} style={styles.itemText} />
          : <Text style={styles.itemText}>{props.title}</Text>
        }
        <View style={styles.btnContainer}>
          <Buttons title="Delete" onPress={() => props.pressHandler(props.todoKey)} style={styles.itemBtn} />
          {isEditing 
            ? <Buttons title="Save" onPress={handleEdit} style={styles.editBtn} />
            : <Buttons title="Edit" onPress={() => setEdit(true)} style={styles.editBtn} />
          }
        </View>
      </View>
    </View>
  );
};
const styles = /* ... */
Here is codesandbox with a code: https://codesandbox.io/s/public-edit-todo-item-bsc9p
EDIT 1
If you want to load current TODO title during edit instead of clearing it first, change <TodoItem />:
- set props.titleas initial value oftext
- remove setText("")fromhandleEdit- it's not needed anymore
const TodoItem = props => {
  const [text, setText] = useState(props.title);
  const [isEditing, setEdit] = useState(false);
  const handleEdit = () => {
    props.editHandler(props.todoKey, text);
    setEdit(false);
  };
  return (
    {/* stays the same */}
  )
}