How can I call a static method from a class if all I have is a string of the class name?

Depending on version of PHP:

call_user_func(array($class_name, 'doSomething'));
call_user_func($class_name .'::doSomething'); // >5.2.3

Use call_user_func. Also read up on PHP callbacks.

call_user_func(array($class_name, 'doSomething'), $arguments);

These answers are all outdated:

<?php

class MyTest{
    
    public static function bippo(){
        echo "hello";
    }
}

$a = MyTest::class;
$a::bippo();

works fine

After I have almost missed the simplest solution from VolkerK, I have decided to extend and put it in a post. This is how to call the static members on the instance class

// calling class static method
$className = get_class($this);
$result = $className::caluclate($arg1, $arg2);

// using class static member
foreach ($className::$fields as $field) {
  :
}

Reflection (PHP 5 supports it) is how you'd do this. Read that page and you should be able to figure out how to invoke the function like that.

$func = new ReflectionFunction('somefunction');
$func->invoke();

Documentation Link

if you need to adjust the namespace

$call = call_user_func(array('\\App\\Models\\'.$class_name, "doSomething"));