Is it permissible to accept and use an anonymous gift, which is a mobile phone placed on the window sill of the children's room, found with a note stating, "A gift for you, so-and-so son of so-and-so"?

The fundamental principle in Islamic law is that a Muslim should not take the property of another except with their consent. If this consent is clearly established through legitimate evidence, then it is permissible to take the property. Among the recognized forms of evidence are reasonable circumstantial indications (Qarā’in), which are any apparent signs that point to something hidden. Circumstantial indications that lead to a strong presumption (Ghalabat al-Dhann) are considered valid proofs upon which rulings can be based.

Scholars have meticulously examined the Islamic Law (Shari'ah) in its sources and resources and found that it recognizes circumstantial indications and bases rulings upon them. An example of this is the Hadith (definition) of Jabir ibn Abdullah (may Allah be pleased with him), when he intended to go to Khaybar. The Prophet (peace be upon him) said to him: "When you go to my agent, take fifteen Wasqs from him. If he asks you for a sign, place your hand on his collarbone."

Therefore, the presence of a mobile phone on the house window, accompanied by a note stating that it is a gift for so-and-so, constitutes circumstantial evidence that indicates with certainty that it is a permissible gift for you, and it becomes the property of your son.