Son of the great Egyptian Jewish singer and composer Ibrahim Zaki Mordechai (known as Zaki Mourad), Mounir was known for, among other things, his efforts to incorporate jazz music into Egyptian popular music.
He wrote the music for one of Shadia's big hits, "Wahed, Etneen" (One, Two), which you should check out. The Hawaiian guitar at the end of the song is just perfect.
Like his sister Leila, Mounir converted to Islam after he married. His wife was the actress Soheir El Bably.
But unlike Leila, it seems that Mounir was not accused of relations with Israel, and so he remained active as an artist at least into the 1960's. His film appearances are quite delightful, and I've found several on youtube. They show him doing fare that is somewhat more "modern" (not sure how else to put it) than that done by his sister, and would seem to put him on the avant-garde side in the Egyptian popular music scene.
Check out this jazzy number, "Ayna Tadhhab Hadha al-Masa'" (Where are you going this evening?) from the 1955 film Appointment with Satan (موعد مع ابليس), which features Mounir on vocals (and he co-wrote the music with Mahmoud El-Sherif).
This is another delightful one, "Isti‘rad al-Batta" (Duck Review -- not really sure how to translate this), from the film Bint al-Hitta (Local Girl, 1964), with Mounir on vocals (and he composed the music). It's also notable for the dancing of the divine, inimitable Samia Gamal.
This long number is really a scream. It's called "Skatch al-Kura (Ya Hadarat al-Mustama‘in)" or Football Sketch (O Distinguished Listeners), sung by Mounir, and again, his composition, and one showing his equal facility with modern show tunes and traditional Egyptian mawwal. It's from the film Naharak Sa‘id (1955), and it's a marvelous musical production, beautifully filmed, by director Fatin Abdel Wahab, who later married Mounir's sister Leila (I'm not sure which year, but definitely after this film was made.
According to the wikipedia article about Mounir Mourad, he wrote the music hits not just for Shadia, but several for Abdel Halim Hafez (including the great "Dahk wa La‘b") as well as hits for the likes of Warda, Muhammad Qandil, Hani Shaker and others.
Finally, download a zany instrumental with insane Farfisa, called "The Factory Theme," here.