Here is a possible solution for a Dart function that randomly generates unique names without using predefined lists of names:

main.dart

import 'dart:math';

// A set to store the generated names.
final Set<String> generatedNames = Set();

String generateUniqueName() {
  // Generate a random first and last name.
  final firstName = generateRandomName();
  final lastName = generateRandomName();

  // Concatenate the first and last name to form a full name.
  final name = "$firstName $lastName";

  // If the name is not in the set of generated names, add it and return it.
  // Otherwise, generate a new name.
  if (!generatedNames.contains(name)) {
    generatedNames.add(name);
    return name;
  } else {
    return generateUniqueName();
  }
}

// Generates a random name by picking a random letter from the alphabet
// and repeating it a random number of times (between 2 and 6).
String generateRandomName() {
  final random = Random();
  final charCode = random.nextInt(26) + 97; // Generate a random letter from 'a' to 'z'.
  final char = String.fromCharCode(charCode);
  final length = random.nextInt(5) + 2; // Generate a random length between 2 and 6.
  return char * length; // Repeat the letter the given number of times.
}