Before I begin, I cannot answer this in full, since I am absolutely no expert. I can however point you in the direction of a circle of scholars (linguists, anthropologists, and cultural historians) that is able to give you more insight in this.

The connection I have within this circle is Dr. Tomáš Petrů, who - as part of this group of scholars - contends that there existed a so-called 'Persosphere' between the 11th and 17th centuries in Central and East Asia. This sphere contained the major Arab empires during the Islamic golden age, the Mughal Empire, Ayutthaya, the Malay sultanates, and even the late Mongol and Timurid khanates. It is argued that the sophisticated status and widespread presence of Persian intelligentsia (if one can call it such in that period) across Asia saw an incredible dissemination of Persian cultural, economic, and administrative practices that was often emulated and merged with local traditions.

Now to get to the specific case of East Africa it is worthwhile looking at the influence that Yemeni and Omani (and even Iranian) traders had in that region. From the Somalia all the way down to Tanzania these traders operated, often from settlements-turned-tradeposts like Zanzibar.

The term 'Zanzibar' itself is of Persian origin, interestingly. Linguists argue that the suffix '-bar' is a linguistic corruption of the Persian word 'bandar', which means 'port' (for example, a place like Bandar-e-Abbas, on the southern Iranian littoral translates to 'Port Abbas').

The source for this not very well-phrased explanation is:

omáš Petrů, PhD.* Islamic Vigilantes in Indonesia: A Serious Threat or a Spent Force?*, EUROSEAS Conference, 11.-14. srpna 2015, Vídeň The Malay World as Part of the Persian Cosmopolis. Intra-Asian Connections: Interactions, Flows, Landscape. 22.-24. října 2014 Asian Dynamics Initiative, University of Copenhagen

I can't find any common answer! the most widely accepted theory is just unsatisfactory because it claims that Persian Muslim traders came to the that region and founded the city in the 10th century and they somehow started ruling it. First of all Mogadishu was older than that and more importantly, there is absolutely no evidence of ever a "Shah" ruling the city. It was either Arabs or "Berbers" (Somalis) because Ibn Batuta said that a man from Berbera who spoke the Berbera language and Arabic ruled the city when he visited it in 1330 when it was under the Ajurans.

Persian influence cannot be left out though! What's absolutely facinating to me is that the greater Mogadishu region celebrates a festival known as "Nayruus iyo Dab Shid" meaning "Nayruus and Fire lighting" where there is fire play and ritual lighting of fire. This festival is celebrated on the New year's day of the ancient Somali solar calendar which begins between July 21-July 23 of every year. On July 23 2015 they were celebrating new years 3141. The Ajurans probably throught it was too Pagan and made up Istunka (combat competitons) to replace the holiday with something less Pagan.

This Nayruus fire ceremony maybe has something to do with Zoroastrianism? 10 century Persian Muslims could not have spread Zoroastrianism!! Maybe it was the Sassanids? or The more likely answer... the Achaemenids because the people started writing this calendar in the 5th century BCE, the time of the Achaemenid empire??

The Horn region, especially Somalia is severely under-studied. First projects were carried out in 2007 and they since found a 5000 year old cave painting, ancient tablets and countless ruins.

So, the Persian etymology is the most commonly offered for the origin of the name.

However, there is an alternate etymology offered, coming from the Swahili phrase Muji wa mwyisho which means "the last city" in the sense of being the northernmost city in the Swahili sphere of influence. Neville Chittick favored a Bantu derivation^1, though he may have been influenced by his career conducting archaeological surveys of Swahili cities. Other supporters of this derivation also pointed to other place names like the Shangani district, Kisimayu and Marika as having Swahili meanings^2

On the other hand, I did just say that the Persian derivation is the most common one. That is because Swahili chronicles from Kilwa and Pate speak to early settlers in Mogadishu being from Shiraz, a Persian province. From Mogadishu, the descendants of the earliest settlers were said to have traveled south and founded cities like Kilwa, Lamu and Pate. The leaders of Kilwa when the Kilwa Chronicle was written (early 1500s) claimed descent from these Shirazi founders.

This lead scholars of the Swahili coast in the 1930s to the 1960s to posit a "shirazi phase" of East African history, where significant immigration from Persia was responsible for the foundations of Swahili culture. In the 1970s and 1980s, this proposition was strongly revised, with revisionist historians pointing to an absence of Persian influence on Swahili language, Architecture, or holidays^3.

Even though scholarly opinion now generally regards Swahili cities and Mogadishu as having been already extant African settlements when Arab and Persian mariners arrived, it is undeniable that there were strong commercial contacts with the Persian gulf. Virtually every archaeological survey of Swahili sites has turned up examples of Iranian pottery and metalware.

1 CHITTICK, NEVILLE. 1982. “MEDIAEVAL MOGADISHU”. Paideuma 28. Frobenius Institute: 45–62. http://www.jstor.org/stable/41409873.Copy

2 Kassim, Mohamed M.. 1995. “Islam and Swahili Culture on the Banadir Coast”. Northeast African Studies 2 (3). Michigan State University Press: 21–37. http://www.jstor.org/stable/41931111.

3 DE V. ALLEN, J.. 1982. “THE "SHIRAZI" PROBLEM IN EAST AFRICAN COASTAL HISTORY”. Paideuma 28. Frobenius Institute: 9–27. http://www.jstor.org/stable/41409871.