Asterisk symbol (*) is used to check values in the array, not the array itself.

$validator = Validator::make($request->all(), [
    "names"    => "required|array|min:3",
    "names.*"  => "required|string|distinct|min:3",
]);

In the example above:

• "names" must be an array with at least 3 elements,
• values in the "names" 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",
]);