A girls name could be anything - they won't have the pressure to pick a "regnal" name, only a "royal" one. I doubt the Queen will let them use their own name as the first name - she's known to be a modest person. Personaly I would love Cecily - utterly English, traditional, and it's a family name, too (that of The Queen's grandmother and Prince Philips's sister). I just hope they don't use Isabella - it's the same name as Elizabeth and it would cause problems with the numbering (Isabella is written "Elisabeth" in Latin; and the Queen is Isabel II in many countries).
I'm convinced a boy will be George. I'd love John - Prince Jack - but I doubt they would be allowed to use it.
They won't use Arthur, Albert, Henry, James, Edward, Charles, Philip, William, Spencer, Anne, Alexandra, Beatrice, Louise, Catherine, or Diana as first names: they're either taken by close family members or are "forbidden" for regnal names. I wish people realised that a Queen Diana would be a slap on the face of the Queen and Prince Philip - and William isn't that close to the Spencers (and Spencer isn't seen as a "proper Christian name", really).
One think I would love to see was a nice Welsh / Cornish / Scottish name included. I loved how the little Danish royal twins have Faroesan middle names.
I think we could have the kid be known publically as a nickname name - like Harry and Kate. I could see a Princess Mary known as Princess May (like The Queen's late grandmother), or a even a Prince William Jr. known as Liam.
Taking in consideration that the little Prince or Princess, though not in the same generation as them, probably will reign at the same time as a lot of today's heirs of heirs (whereas he or she will be the heir of the heir of the heir), to avoid confusion (that goes for girls names in particular, seeing as Prince Christian of Denmark currently is the only boy heir in that generation) they might avoid names like Elizabeth (Princess Elisabeth of Belgium), Catherine or Amelia (Princess Catharina-Amalia of the Netherlands), Eleanor (Infanta Leonor of Spain) or Christian (the before-mentioned Prince Christian of Denmark).
As I posted in the other thread...
Diana would not be allowed as the throne has to approve the baby's name (since she may one day be queen) and Queen Elizabeth didn't like Princess Diana.
I think William would still want to honour his mother, so my suggestion is Frances (Diana’s MN).
My pick is…
Victoria Catherine Elizabeth Frances
For a boy:
I think Philip (after Wills Grandfather) is a likely contender and Arthur (Wills second name). Most likely Charles after his father and though it’s unlikely, I think Spencer after his mother (her madien name) is a great option.
Philip Arthur Spencer Charles