How to set a default value for an optional positional parameter of type Function?

You can only use constant values (aka. compile-time constants) as default values. You cannot create a constant function literal, so there is no way to write the function in-line in the constructor.

However, references to top-level or static functions are constants, so you can declare the default value function as a static function or top-level function.

void main() {
  Person p = Person();
  print(p.foo('Hello')); // Prints "Hello"
}

class Person {
  final String Function(String) foo;
  Person({this.foo = _identity});
  static String _identity(String value) => value;
}
// or as top-level.
// String _identity(String value) => value;

You can (and should) choose to make the function public if the default value is on an instance method, and you expect anyone to extend or implement your class. In that case, they need to declare the same default value.

Another option, which is often at least as useful, is to not use a default value, but replace a null before using the value:

class Person {
  final String Function(String) foo;
  Person({String Function(String) foo}) : foo = foo ?? _identity;
  static String _identity(String value) => value;
}

or even using a non-constant value:

class Person {
  final String Function(String) foo;
  Person({String Function(String) foo}) : foo = (foo ?? (String x) => x);
}

For a constructor, it makes very little difference. If it was an instance method instead, using ?? to replace null avoids subclasses having to use the exact same function as default value.

Personally I recommend always using ?? instead of a default value. It's more flexible since it allows non-constant values. For non-function default values, you'll have to document the default behavior instead of just letting the dartDoc show {int x = 42}, but for functions, you'll have to document them anyway.