It stands to reason that Julia's full name is probably Julia Vaughan, but The Hours never makes this perfectly clear.
Julia is Clarissa Vaughan's teenaged daughter, and at eighteen or nineteen years old, she seems to predate Clarissa and Sally's relationship. We know that Julia is at least eighteen, because Clarissa mentions the ring that she gave to Julia on her eighteenth birthday (14.7). We also know that Clarissa and Sally have not yet celebrated their eighteenth anniversary (11.110). Hmm.
The short of it is that we don't really know if Clarissa decided to have a kid before or after she met Sally, but we do know that Sally and Clarissa raised Julia together.
Does Julia resent growing up with two mothers and no father? Clarissa worries that she does (14.22), but that seems to be a product of Mom Anxiety more so than an accurate perception of reality. After all, Julia is now "in thrall" to a queer theorist who's set on disrupting gender norms (1.31), and something tells us that if she was ever frustrated by her upbringing, she's well over it now.
When Louis Waters looks at Julia, he sees a young woman who "isn't beautiful," but who is instead "handsome and assured in the way of a young athlete, her head all but shaved, her skin pink" (11.126). Here's how the narrator of the novel describes her:
She is lush and strong, crackling with health, like some kind of idealized Irish farm girl just in from the fields. She must take after her father (Louis has fantasized about him, imagined him as a strapping young blond, hard up, an actor or painter maybe, a lover, a criminal, a desperate boy, down to selling his fluids, blood to the blood bank and sperm to the sperm bank). He must, Louis thinks, have been huge, rugged, a figure of Celtic myth, for here now is Julia, who even in her tank top and shorts, her black combat boots, looks as though she should be carrying a sheaf of barley under one arm and a new lamb under the other. (11.128)
It's clear that Julia takes after her queer theorist mentor, Mary Krull, who also favors short-cropped hair and combat boots. But, for all of her interest in cutting-edge gender politics, there's also something pretty old-fashioned about Julia, so much so that we might even think of her as an "old soul." Julia says "goodbye" and "hello" rather than "hi" and "bye," and at times, Clarissa has the impression that her daughter is much older than she seems.
Take a gander at this passage, for instance: "Julia sighs with a surprisingly elderly mixture of rue and exhausted patience, and she seems, briefly, like a figure of ancient maternal remonstrance; part of a centuries-long line of women who have sighed with rue and exhausted patience over the strange passions of men" (14.2).
Or, get a load of this one: "Beside a vase full of roses, Julia sleeps on the sofa with a book open on her lap. In sleep she sits with an air of surprising dignity, even authority, foursquare, shoulders relaxed and both feet on the floor, head bowed discreetly forward, as if in prayer. At this moment she could be a minor goddess come to attend to mortal anxiety; come to sit with grave, loving certainty and whisper, from her trance, to those enter, It's all right, don't be frightened, all you have to do is die" (22.5).
Why do you think Julia is described in this way? What is it about her or her upbringing that makes her seem like an old soul? Maybe Julia can be seen as a continuation of the strong women we see throughout the novel: Virginia, Laura, Clarissa herself.