Might just be thrown together, like this week.
(This is not really the time or place to go into it, but there is a substantive case to be made that Ethiopia is one lobe -- the other being "Arabia Felix/Yemen" -- of an almost unthinkably under-rated civilization, probably Jewish at its core (ca 800 BC-350 AD), that played a significant role in at the very least commercial history from the 14th century BC to 632 and the Arab unification, which was itself probably made possible by this civilization's previous commercial (and cultural?) unification of Arabia. This is not an AH about eliminating that civilization.)
(There is also a glorious tradition in SF of positing Ethiopia or a country clearly based on Ethiopia (often riffing on Marcus Garvey's millennial visions) as the center of a large, independent African empire rising in good SFnal fashion amid plenty of lectures about the author's notion of politics. Cf. Black Empire by George Schuyler, and the "North Africa Trilogy" by Mack Reynolds beginning with Black Man's Burden. This is not an AH about what if one of those happened, in 1920 or 1960.)
(You may be thinking that I'm throwing away some good ideas. I may be thinking that, too.)
So I'm (re) reading George Macdonald Fraser's Flashman on the March, which puts the great rogue and poltroon in the heart of the British invasion of Abyssinia in 1868, at that time under the murderous Emperor Tewodros II. Wikipedia doesn't dwell overmuch on the murderous angle, so much as the "unifier of modern Ethiopia" angle.
So what if he hadn't ever come to power, or had gone just a little crazier a little earlier, and fissioned the Abyssinian states for good in the 1850s? In OTL, Britain took over Aden in 1838, but left Somaliland until 1888 (when the Italians started in). The French take Djibouti in 1883. Italy took Eritrea in 1890, as a jumping off point for invading Abyssinia in 1895. They lost, not least because the Emperor Menelik was able to build on the nascent nationalism created by Tewodoros (mostly by slaughtering all other likely contenders for central authority), and because Menelik was able to play the other European powers off against each other to accumulate modern weaponry.
Let's postulate that the Italians are still the only Europeans interested in conquering the godforsaken, not-very-strategic Abyssinian interior, with the tacit cooperation (as they had in OTL) of France and Britain, interested in securing Italian support in North Africa and against Bismarck's burgeoning Tripartite Alliance (Russia, Austria, Germany). In the AH, Italy begins the piecemeal conquest of the Abyssinian states about when it did in OTL, with the acquisition of Aseb in 1882. It takes them a decade or two, but by 1907 (OTL's establishment of Italian Somaliland), there is a broad "Italian Raj" in East Africa.
Which the British take away from them in 1941, just as they did in OTL. The British take it over as a UN Trust Territory after the war, just as they did with Eritrea (until 1951) and Italian Somaliland (until 1949), and then turn it back over to the Italians (again, just like Eritrea and Somaliland in OTL), and it becomes independent as a giant Ethiopian Union in 1960 -- and immediately comes apart in civil war, with the British or Italians or French fighting against the Communist forces (just like in Aden in OTL) and Nasser putting his oar in, and the whole sorry megillah. Post-colonial history tends oddly toward the centrifugal; it's possible that the final outcome (ca. 1966) is a repressive Ethiopian Union (no monarchy for our brave new Ethiopia) with a slow-burning set of Communist rebels in Somaliland, Tigre, and elsewhere. Maybe it liberalizes somewhat post-1989, as South Korea and other such places did. Unless the Communists win the war, there's no terror-famine, no knock-on Somali War and famines ... generally things are rather better than OTL, though you'd have a hard time convincing the people in the Union's prisons of that.
Or maybe it goes Communist (or Soviet-dominated anyhow) in the 1970s, just like OTL Ethiopia and Somalia, and then implodes messily in the 1990s to combine the best features of Bosnia, Somalia, and Afghanistan in OTL.
Oddly, the biggest impact from our perspective in OTL is the absence of Rastafarianism, which depended rather a lot on the heroic and independent image of Ras Tafari, the Emperor of Ethiopia better known as Haile Selassie. What Jamaican Garveyism becomes without Selassie, I don't know -- weird Sufi-style Islam worshipping Mohammed V of Morocco? Does Bob Marley make rai or gnawa famous by blending it with ska?
What does this do gaming-wise? It gives you a place to set Cold War guerrilla wars where nobody knows the outcome, without affecting what year the M4 carbine or the AKM is introduced, or even who's President or Premier. Like a lot of "close parallels," its advantage is the frisson of unfamiliarity, but with the ease of the normal.