No details are given about the boy's age or how he died.
However, Saudi news website Al Arabiya said sources had revealed he was 12 and suggested he had died of an illness.
The note says that young Osama often play-acted scenes of martyrdom while playing with other children - but it is not known where he lived.
In November, Osama bin Laden's "favourite" son called on Muslims around the world to launch terror attacks on the US for killing his father.
In an audio message, Hamza bin Laden, believed to be around 28, urged Muslims to "to take revenge on the Americans, the murderers of the Shaykh [Osama bin Laden], specifically on those who participated in this heinous crime."
It came just days after Hamza's wedding video was released, showing the "Crown Prince of Terror" as an adult for the first time.
This was released by the CIA as part of a trove of material recovered during the May 2011 raid that killed the terror group leader at his compound in Pakistan.
Until then, the public had only seen childhood photos of Hamza, which have been used as propaganda tools by Al-Qaeda. It's believed the militants have not released pictures of him as an adult for his own safety.
In recent years, he has emerged as the heir to his father's throne and become a prominent member of the terrorist organisation.
Dubbed the "Crown Prince of Terror", he has purportedly called for terrorist attacks on London, Washington and Paris in an audio message released by Al-Qaeda.
In January of this year, he was officially designated a terrorist by the United States.
The U.S. State Department added him, to its list of global terrorists after confirming he had followed in his father's footsteps to join Al-Qaeda.
Al-Qaeda senior leader Ayman al-Zawahiri released a tape in August 2015, claiming that Hamza had joined the group. In the audio message, Hamza himself spoke and called for lone wolf attacks.
The Al Qaeda leadership believes there is huge propaganda value in invoking the bin Laden name because it allows them to stake their claim to be the legitimate representative of jihad in the Middle East.
In recent years Al-Qaeda and its Syrian affiliate, Jabhat al Nusra, have lost territory and fighters to the more dominant ISIS during a long-running mutually destructive conflict. Al-Qaeda has also lost key leaders as a result of ISIS assassinations and Coalition bombings.