Wa `alaykum as-Salamu wa Rahmatullahi wa Barakat.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
It is highly recommended to make du`aa on Friday specially after Asr Prayer.
In his famous work, Fiqh As-Sunnah, Sheikh Sayyid Sabiq, elaborates on the subject as follows:
Duaa on Friday
A Muslim should do his best to make duaa during the last moments (or hours) of Friday. `Abdullah ibn Salam relates, “I said, while the Messenger of Allah (peace and blessings of Allah be upon him) was sitting: ‘We find in the Book of Allah that on Friday there is an ‘hour’ in which, if a believing slave supplicates to Allah for something, his supplication (du`aa’) is (indeed) accepted and he is granted whatever he prays for.’ The Messenger of Allah (peace and blessings of Allah be upon him) pointed toward me and said, ‘Or part of an hour.’ I said. ‘You have spoken the truth; or part of an hour.’ I asked, ‘Which hour is it?’ He replied, ‘The last hour of the day.‘ I remarked, ‘That is not a time of salah (Prayer)?’ He responded, ‘Certainly [it is]; if a believing slave offers Prayer and then sits, he will not be sitting, save due to the Prayer, and he will be in Prayer.'” (Ibn Majah)
Abu Sa`id Al-Khudri and Abu Hurairah (may Allah be pleased with them) report that the Messenger of Allah (peace and blessings of Allah be upon him) said, “On Jumu`ah, there is a time that if a believing slave asks Allah during it for some good, [Allah will definitely] give it to him, and that time is after the `Asr Prayer.” (Ahmad)
Jabir ibn `Abdullah (may Allah be pleased with him) reports that the Messenger of Allah (peace and blessings of Allah be upon him) said, “The day of Jumu`ah has twelve hours, and during one of the hours, you will not find a Muslim slave [of Allah] asking Allah for something, but that He will give it to him. Seek it in the last hour after `Asr Prayer.” (An-Nasa’i, Abu Dawud, and by al-Hakim)
Allah Almighty knows best.