This is a popular question put forth by Atheists who want to challenge believers. But before trying to answer the question, we should first examine its logical validity. Because only logically valid questions are qualified for answering.
Now by the very concept, we know that God is the greatest, the most powerful, the all-dominant existence; in a way that nothing can exist independent of Him or out of His reign, let alone being weightier than Him or in anyway exceeding Him. So the question is just logically absurd to begin with, because there's nothing weightier than (or in any respect exceeding) Allah that can be imagined, and thus a logically absurd concept can never be subject to the power of God because it just cannot exist. So if a thing is impossible to exist due to a logical necessity it is completely out of question of Allah's power or capability. In short, logically we just cannot even begin to answer such a question.
PS: Noteworthy, I think, the question stems from a failure in understanding the difference between logic as a normative science of knowledge and philosophy as a science of existence. Logic is to tell us how existence should be examined as it is, by setting rules for preventing false statements about the reality. So when something is logically impossible that means it essentially cannot exist in the world as it is. But it doesn't necessarily indicate an incapability on the part of the subject under philosophical examination.