There is no satisfactory answer to this question, unfortunately. Some families simply did not have a cognomen, and we don't know enough about the mysterious origins of the cognomen to understand why precisely. There are three distinct types: some were apparently ancient, possibly as old as the 5th century BCE or earlier, and frequently associated with a derogatory reference to a body part (Rufus = "red"; Ahenobarbus = "bronze-bearded"; Cicero = "chick-pea"; Caesar = "curly"; etc). There are many many theories about the strange origins of this nicknaming practice, but most agree that they were used to mark out aristocratic status. Why they were derogatory is another matter, and this seems to go against everything we know about Indo-European naming conventions. These "bodypart" cognomina were hereditary to all male offspring.

A second type was that bestowed upon a Roman for a successful (usually martial) deed, like "Britannicus" or "Africanus" or "Numidicus." These were sometimes hereditary, sometimes not, and sometimes only for the senior son. Unlike the above, these were not ancient, and were often self-imposed. So too was the third group, which (apparently) were also somehow derogatory. Imagine unpleasant nicknames soldiers give to one another on the march. This last group probably aren't properly cognomina in the aristocratic sense.

In general, the cognomen was reserved for the nobiles, or the senatorial class. Our Marcus Antonius, however, was from the plebeian branch of the gens Antonia, which never appears to have any cognomen associated with it. The Patrician branch of the gens Antonia was associated with the cognomen Merenda, and also tended to use different praenomens (I think they used Titus versus Marcus, if I recall correctly). This name cognomen Merenda was frequently omitted even in the Late Republic, however.

There are some other notable families, many of great antiquity, who never seemed to use a cognomen: the gens Hortensia often had none, though the famous Orator and opponent of Cicero is reported to have the tria nomina of Quintus Hortensius Hortalus (no explanation). Others which often appear(ed) without cognomina are Herrenius, Vargunteius, and Vatinius, all with accomplished members.