He received a gift from an unknown person

One of the established principles in Islamic teachings is that the Muslim should not take something that belongs someone else unless it is given with the owner’s willing consent. So if it becomes clear that there was consent, on the basis of acceptable proof, then it is permissible to take that thing in that case.

Proof that is regarded as acceptable, according to the purists, includes reasonable circumstantial evidence.

Shaykh az-Zarqa (may Allah have mercy on him) said:

What is meant by circumstantial evidence is any clear sign that clarifies a subtle issue and confirms it."(Al-Madkhal al-Fiqhi al-‘Am  2/936).

Circumstantial evidence that is that which makes an issue go from being possible to being most likely to be the case, and what it indicates is to be taken into account as being reliable, unless it is contradicted by other evidence that is stronger than it.

Ibn al-Qayyim (may Allah have mercy on him) said:

The Lawgiver did not disregard signs, indications and circumstantial evidence; rather whoever examines the religious texts will find proof that these things carry weight and rulings may be based on them."(At-Turuq al-Hukmiyyah  1/27).

The scholars have a great deal of evidence to support that, including the report narrated by Abu Dawud (3632) from Jabir ibn ‘Abdillah, who said: I wanted to go out to Khaybar, so I came to the Messenger of Allah (blessings and peace of Allah be upon him) and greeted him with salaam, and I said to him: I want to go out to Khaybar. He said: “When you come to my deputy, take five wasqs from him. If he asks you for a sign, place your hand on his collar bone.”

The isnad of this hadith was classed as hasan by al-Hafiz Ibn Hajar, who said: al-Bukhari narrated part of it in a mu‘allaq report at the end of Kitab al-Khumus."(At-Talkhis al-Habir  3/122).

Based on that:

The fact that you found a mobile phone in the window of the house, accompanied by a message stating that its owner was giving it as a gift to So-and-so, is circumstantial evidence which indicates that it is a gift and is permissible for you, and that as a result it becomes the property of your son.

And Allah knows best.