Just because someone is a competent statesman (or musician, or athlete, or writer, or whatever), does not mean their offspring will automatically inherit these qualities. My answer will not be about the specific circumstances of Marcus Aurelius and Commodus, but more about the general idea of the value of ancestry as legitimacy for ruling, and the way this has evolved in history. Instead of focussing on the specific historical circumstances, this answer will be more oriented to examine the possible assumptions behind this question.

To be honest, it's actually a little surprising to me to see this question. It implies that a good emperor or king will, as a rule, produce good successors. It seems like this line of thought is a little out of time. For instance, in the middle ages, being born in a certain family did actually legitimise the social position you were in. The son of a farmer would most likely become a farmer, and the son of a king would most likely become a king. In any case, the farmer's son was hardly a candidate for kingship. Peasants were generally not even considered for a role in governing. The idea itself would be seen as preposterous, especially amongst the ruling class. As a result, there was little room for social mobility compared to modern western society.

This kind of hereditary determinism came under heavy criticism at the end of the middle ages. And rightly so. Sons of capable kings could easily turn out to be bad rulers or just plain crazy. Furthermore, basing the social position of someone solely based on their ancestry is today seen as unfair and even immoral. People should be judged on their own actions and merit, and most of today's western society is based on the idea that there has to be at least some levelling of the playing field, so all people have somewhat a fair chance at making their own lives.

This is not just a random ideological standpoint. The fact that the majority of our current day politicians, leaders of industry, intellectuals and other prominent members of society are not descendants of noble families is, I think, proof of this. Examples of rulers from noble families that turned out, let's say less than ideal, are also plenty (Hippias, Caligula, Ivan the Terrible and George III are some that come to mind).

The French Revolution, American Revolution, the ideas of the Enlightenment and modern democratic ideals can be seen partly as a reaction to the archaic idea that noble families produce the best people (aristocracy literally means 'the rule of the best'). People's capabilities are not determined by their genetical background.

Another train of thought might be that it is less the genetic background, and more the upbringing that one gets that is important to their capabilities in later life. I think there is much to be said for this. Nevertheless this does not mean that good rulers necessarily are good parents. Growing up in the protected and elite world of a royal family could just as easily make someone arrogant, egotistic and devoid of a connection with the world of the subjects he or she is to govern.

Criticism on the idea that good rulers produce good successors is not just an early modern idea. The ancient Greeks already entertained this idea (even though their societies largely remained aristocratical). Both Herodotus and Aristotle praise the tiran Peisistratos, but clearly state his son Hippias did not posses the good qualities of his father. Over 2000 years ago, the Greeks already had the idea that absolute rulers should be avoided, since that much power will eventually corrupt, especially for successors who are born into their powerful position.

Unfortunately, I cannot give a detailed response to the specific case of Commodus. I think the reason he turned out less of a capable ruler than his father will have many reasons and causes, the majority of which we cannot possible know. Explanations would necessarily be highly speculative, since we have no direct and extensive access to the psychology of Commodus. Hopefully someone more knowledgeable about Marcus Aurelius and Commodus can chime in here.

Until that happens, my more general answer is: Commodus turned out incompetent because inheriting a position of enormous power simply because you were born into the right family is not only a bad indicator of competence, but can actually be very detrimental.