bar Cochba appears in rabbinic literature. Josephus does not.

Josephus does not get a mention in Judean text/Talmudic literature.

Simon bar Cochba des, under the name Simon ben Coziba  כוזיבה (his real name was Simon ben Coseba  כוסבה). 

Coseba is the name of a city in Judea.  It has been presumed that “bar Cosiba” means that he was from that city.  “bar,”  however, like “ben” means “son of” a person, not “resident of “ a city.

It is probable that even his “real” name was intended as a device to conceal who he was to protect his family from the possibility of reprisals.

It is claimed that rabbinic authors made reference to bar Cochba as “Cozeba” because that means “man of lies” or “man of deception.”  If “bar Coseba” does not mean “man of the city of Coseba,”  “bar Cozeba” cannot mean “man of lies.”  If we were being consistent in our translation (always a good thing), it would mean “son of lies” or “son of deception.”  That would make a different kind of problem, because it would mean that the rabbinic sources were denouncing the father as a liar, not the son.

The reference to bar Cochba appears in Lamentations Rabbah.  This should not be taken as a historical validation of anything except that the early Rabbis remembered him in the list of martyrs.  The interpretation that “bar kozeba” means “son of lies” is more likely to be a reflection of great grief that the revolt, which began well and resulted for a time in the reinstitution of the sovereignty of Israel, ended in the near genocide of the people of Judea.  Someone has to take the blame for the failure of the revolt, and the leader is the one most likely to be assigned blame.

More than that, however, it is important to note that rabbinic authors do reference bar Cocbha, by whatever name.  While Josephus, the man who claimed a noble pedigree and exalted social status, is not mentioned by them at all.