After much recent speculation the Duchess of Cambridge (Catherine of Britain) has gone into labour and been taken to St Mary's Hospital in Paddington, London. Following the recent changes to the succession laws the child will be heir to the throne (or, at the moment, the heir apparent to the heir apparent to the heir apparent) regardless of whether it is a boy or a girl. Thus it seems a future British monarch will be born today (as it is my birthday I naturally think it a good choice of date!)
The child will be a Royal Highness and a Prince(ss) of the United Kingdom of Great Britain and Northern Ireland (and styled Prince(ss) X of Cambridge within Britain). As for names, it would be expected that they would choose one with a British royal history, preferably one already borne by a British monarch (but if the Duke and Duchess of Cambridge have as little sense of history as their Swedish counterparts they may of course choose a foreign name without royal roots, but I somehow feel Queen Elizabeth or Prince Charles will put a stop to any such ideas).
For a girl the most likely names are perhaps Elizabeth or Victoria, with Anne and Mary as other options. Personally I also think Charlotte would be a good choice, it being the name of the "lost" Queen of Britain, i.e. the only child of the future George IV, who died in childbirth in 1817, so that the crown eventually passed to her cousin Victoria, who was born in 1819. Charlotte would also commemorate two of the baby's grandparents, Prince Charles and Carole Middleton.
If it is a boy, there is a wider range of choices, as Britain has had more male than female monarchs. George, James, Alexander, Edward, Henry and Richard are some of the options.