Asterisk symbol (*) means that you want to check VALUES in the array, not the actual array.
$validator = Validator::make($request->all(), [
"name" => "required|array|min:3",
"name.*" => "required|string|distinct|min:3",
]);
In the example above:
"Name" must be an array with at least 3 elements.
Values in the "name" array must be distinct (unique) strings, at least 3 characters long.
EDIT: Since Laravel 5.5 you can call validate() method directly on Request object like so:
$data = $request->validate([
"name" => "required|array|min:3",
"name.*" => "required|string|distinct|min:3",
]);