I'm fairly sure that it is never explicitly stated in an ancient source that the strategoi had to be of noble birth, and in fact this concept was quite alien to the Democracy. BUT: the position of strategos, as you know, was one of the very few which was not chosen by lot in the Democracy. The ten generals were "elected," and that process carried with it all the basic parameters of elections which recur throughout history: money and notoriety get you ahead. For the vast majority of attested strategoi in the 5th and 4th centuries at Athens, we have zero socio-economic information. That makes it difficult to make blanket statements about their identities. It is difficult to draw conclusions for those about which we know more, like Perikles. He held the position of strategos almost continually during his time, and that indicates that the office of general was one possible avenue of holding and wielding political influence under the parameters of the Democracy.

I agree with scholars who assert that the strategoi were, for the most part, probably influential members of Athenian society, either because they were born in to old aristocratic families, or because they were rich, or because they were notorious for other reasons. Sinclair (Democracy and Participation in Athens) has some nice, concise comments on this topic:

By the late fifth century and early fourth, then, it was clear that good birth was not necessary for an ambitious Athenian youth. But was Perikles right in claiming that poverty was not a bar? About the socio-economic position of many Athenians who appear on the scene in the fifth century and the fourth, whether as arkhons, strategoi or other officials, or as speakers in the assembly, we have no information. But for those about whom we have some information it would seem to be generally true that a measure of wealth, whether inherited or acquired, was necessary for those who aspired to leadership, both in the fifth century and in the fourth. Some have gone so far as to argue that in the fourth century political leaders were in terms of socio-economic origins drawn from the middle or propertied classes and formed a coherent social class. Certainly there were leaders with substantial inherited wealth. Timotheos inherited not less than 17 talents from his father Konon, and Demosthenes should have received through his guardians an inheritance of not less than 10 talents. A larger number are known to have belonged to the trierarchic class and came from families of moderate wealth. Such were Thrasyboulos of Steiria, who took a leading part in restoring democracy in 403/2 and who exercised considerable influence in Athens during the next fifteen years, and Kallistratos whose oratorical ability Demosthenes praised and whose financial expertise may be discerned in the 370s and 360s.

But among political leaders there were some who came from poor backgrounds. Demosthenes is far from a reliable witness for his rival Aiskhines, but there may be a grain of truth in the orator's claim that Aiskhines was a rich man who had been a beggar. He is said later to have inherited 5 talents from his brother-in-law and to have acquired a great deal of wealth through political bribes. Though some doubt may remain about Aiskhines' poverty and the extent of his later wealth, no doubt attaches to Demades who came from a poor family but acquired a considerable fortune, mainly through political bribes.

What seems to emerge, then, is that whatever a man's socio-economic origins, an Athenian had to have inherited or himself acquired resources of some extent in order to sustain political ambitions. A similar situation seems to have pertained with those who tended to concentrate their energies on military activity. A background of a well-to-do family or a family with experience in the strategia was undoubtedly an advantage. Khabrias, the victor in the decisive naval battle of Naxos in 376, seems to have been one of those whose father had been wealthy enough to be called on to undertake the trierarchy. Iphikrates on the other hand was the son of a cobbler and was certainly self-conscious about his social origins, but the material rewards gained from his successes as a military commander, whether in Athenian service or not, must have been considerable.