"Have you thought of a name for her?" she asked Cornelia.
"If it was a boy, I was going to name him after his father, Julius."
Oh dear. Let's tackle the error Iggulden ought not to have made. During this period, male Romans were known by three names. As an example, Caesar's name was Gaius Julius Caesar. The first name, Gaius, was used by family and friends. The second name, Julius was the name of the family or clan, the Julii. The final name, Caesar, was used to differentiate between between all the Gaius Julii, as the limited number of first names meant that in large families there could be many men with the same name.
So if he were a boy, as the first born his name would be (Something) Julius Caesar; as Caesar's father and grandfather were called Gaius, there's good odds that he might be given that first name.
But she is a girl. So what will she be called? Well, officially, she doesn't get a name of her own; instead she gets her father's name and, possibly, an indication of birth order. In other words her name is Julia, and if she has any sisters, they'll be known as Julia as well. She may well gain a family nickname, but Cornelia (daughter of Cornelius) doesn't get to name her.
I have one final nit to pick; that volume is called The Death of Kings, but, as far as I can see, only one king actually dies in it.
 Until she's married when she make add or take her husband's name. I will note that during the period there were many exceptions and notable females were often given second names, the equivalent of male third names, to indicate which female they were.
 This has caused some confusion amongst historians between notable sisters.