Servius identifies the “Julius Caesar” named in Aeneid 1.286-8 as the dictator Gaius Julius Caesar rather than as Augustus (a debate still ongoing at present). He then offers two possibilities for the cognomen Caesar:
Caesar vel quod caeso matris ventre natus est, vel quod avus eius in Africa manu propria occidit elephantem, qui caesa dicitur lingua Poenorum.
I seem to recall reading somewhere that birth by Caesarian section was not used in antiquity unless the mother had already died before the procedure was undertaken, and so (if that is correct) that would not have been the case for Julius Caesar himself. The second possibility is interesting: that caesa is the Punic word for “elephant,” and his forebear had killed one in Africa.