I'm really excited to find out the name!
I expect it will be something really really traditional.
Seeing as the child will be placed after Prince William in the line of succession and therefore will be an heir, it's almost 100% certain that his/her name will be top-traditional.
First name: They could choose to go with the ultra traditional names like George, Richard or Edward; Elizabeth, Victoria or Mary. Or they could go with more uncommon yet traditional names like Arthur, Philip, Alexander or Albert; Charlotte, Matilda, Alexandra or Margaret. My money are on either George or Arthur, or Elizabeth, Margaret or Victoria.
"Middle names": As with (almost) every royal baby, the secondary names are probably going to get used to honour family members. A strong contender in my opinion is Francis/Frances, in a girls name being a replacement for Diana. Frances was Diana's middle name and Francis is Michael Middleton's middle name. Francis furthermore has a royal connection in that the father of Queen Mary, the grandmother of Her Majesty the Queen, was Francis, the Duke of Teck. Family names used could be, for a boy: Charles, Louis, Michael, James, Arthur or Philip, for a girl: Elizabeth, Margaret, Anne, Alice, Diana, Alexandra, Carole/Caroline or Mary.
My (first) guesses for a boy:
George Charles Arthur Philip
Arthur Charles Philip Francis
George Philip Arthur Michael
Alexander Philip Arthur George (long-shot)
Albert Michael Charles Philip (long-shot)
My (first) guesses for a girl:
Elizabeth Alice Caroline Frances
Elizabeth Carole Mary Diana
Margaret Caroline Frances Elizabeth
Victoria Carole Diana Elizabeth
Alexandra Elizabeth Caroline Frances (long-shot)
Mary Elizabeth Carole Diana (long-shot)
Alice Diana Elizabeth Carole (long-shot)
I've gotta say, I kind of hope they have a baby girl. I think that the law that boys are first in line even if they have older sisters was recently struck down, or is (hopefully) going to be. That would be great, especially since William and Kate's kid will follow two kings (Charles and William).
Warning: I'm also just geeking out over this. My mom and sister and I made a huge deal out of William and Kate's wedding, just because, politics aside, it's FUN. They seem like nice people. Wish them all the best.
I've gotta say that I kind of hope they don't go with Elizabeth for the first name, as it is a little "done". Victoria's very pretty.
Now, I'd like to take a look at the names of Queen Victoria's kids, as she did have nine to choose from.
Victoria Adelaide Mary Louisa, the Princess Royal: Adelaide and Louisa would be interesting, if unlikely. Adelaide was also likely due to her father, Prince Albert, being German. William and Kate don't have to worry about honoring other nations.
Albert Edward: As much as I love Prince Albert, the name is very egg-head-ish to me. And Edward has a somewhat negative connotation today...
Alice Maud Mary: Alice is lovely, Maud is dowdy, Mary is done. I think Alice could be a strong possibility.
Alfred Ernest Albert: ... I don't like any of these names. But I'm pickier with boy names than girl names.
Helena Augusta Victoria: As unlikely as I find the first two names, I love them! Helena is gorgeous--it means beautiful--and Augusta is very classic and regal.
Louise Caroline Alberta: Not gonna lie, I'd die if they had a kid named Caroline. Because, you know. It's my name. Unlikely though.
Arthur William Patrick Albert: Patrick will not happen, but it's a great name. Arthur seems more likely, and I love it, and I would die, because--King Arthur. Come on. That is too great.
Leopold George Duncan Albert: Leopold would be great, as it has the cool, modern nickname of Leo. Duncan's a cool name, but I can't see it on a prince.
Beatrice Mary Victoria Feodore: There's a none-too-popular Princess Beatrice around right now, so that won't happen. None of these--aside from the obvious Victoria--seem possible.
Arthur is by far my favorite for a boy, aside from the cool but unlikely Alexander. Maybe I just want them to have a girl so they have more name options. (Helena and Augusta are sentimental favorites but I highly doubt either would happen.) (Caroline. Come on. Caroline for the win.)
Edit: I should also add that royals have a history of calling people by names other than their first names (as we saw in The King's Speech) whether they be nicknames or middle names, so who knows what the kid will actually be called. I think it's kind of cool. Also, the name Alexandra (granddaughter of Victoria) is a great name, but if you actually look for the most famous royal the name was given to... It's Alexandra Romanov. The last Tsaritsa of Russia. Who died, as we all know, horribly. So while it's a beautiful name, another royal Alexandra might have a bit of gloom to it for some. I don't mind it, as the Tsaritsa was a fascinating, good woman.
Edited for privacy.
Just throwing in some of the names of the relatives of the little future heir:
Elizabeth Alexandra Mary and Philip
- Charles Philip Arthur George
-- William Arthur Philip Louis
-- Henry Charles Albert David
- Anne Elizabeth Alice Louise
-- Peter Mark Andrew
-- Zara Anne Elizabeth
- Andrew Albert Christian Edward
-- Beatrice Elizabeth Mary
-- Eugenie Victoria Helena
- Edward Antony Richard Louis
-- Louise Alice Elizabeth Mary
-- James Alexander Philip Theo
Michael Francis and Carole Elizabeth
- Catherine Elizabeth
- Philippa Charlotte
- James William