The Byzantine army is not really my area of expertise, but recently I read Justinian's Men by David Parnell, and that was a really good book, I would highly recommend it if this topic interests you.

The strategoi were absolutely appointed directly by the emperor, as was the case since the first Roman emperor, Augustus. The emperors were the commander in chief of the armies, which meant taking a direct hand in military affairs. Naturally, this meant there could be a significant amount of favoritism and nepotism, but maybe not as much as we would expect. The Roman and Byzantine armies placed significant emphasis on military quality and capability. The men who ended up as strategoi often climbed through the ranks, and they were often promoted after some sort of victory or success, and removed from service if they were defeated or failed in their duties.

These positions were not inheritable, and Parnell writes that there is little evidence to suggest having a family member in the army placed made a person more likely to be promoted. But he also says there is a lot of evidence that multiple members of a family might serve concurrently and across generations, and that certain families gained a reputation for military service.