It would be nice to see the BRF include a (non-Greek) royal as a godparent, maybe a Norwegian royal (like Haakon) since they seem to be most-closely related to the BRF of the reigning royal families. I could see them not going with any foreign royalty (Greek or otherwise) at all, though.
Anyway, here's what I would like to see as a *first* name ranked from what I personally like the most to what I like the least, the child's future regnal number I'll also comment on the chances.
Boy:
Richard (I don't see it as too conflicting with the other Prince Richard, their lifespans are unlikely to overlap that much. But it's a longshot. I like it though.)
Charles (could be known as "Charlie" or something)
William (don't know the likelihood of repeating names so closely anymore)
Henry (Don't think it would present too much confusion since the other Henry, as we all know, is Harry
)
Edward (...but I just don't see it happening with two Prince Edwards)
Philip (The only issue I could see is a numbering issue for the future monarch. Would they count Mary I's husband, since he was "King"?)
Michael (Real longshot but it's not a bad name and pretty traditional, maybe not for the BRF, but royalty in general)
George (Most likely, IMO, but probably my least favorite of the "traditional" names)
Girl:
Elizabeth (Obvious choice, and if the baby is female, my #1 choice)
Anne (second favorite choice, Anne II has a nice ring to it)
Victoria (I don't know if they would "go there", but I don't think QV would have been that opposed, after all, she named her eldest child Victoria, and she could have ended up queen if QV stopped at one
)
Charlotte (could have negative connotations, but it's a nice name and would be a departure from traditional female British monarch names. Would also honor Charles)
Caroline (another nice name that would honor Carole Middleton as well as Charles)
Catherine (not a "traditional" BRF name, however, it sounds quite royal and would be different enough from "Kate" to work. I believe the child would also be a descendant of Catherine the Great.)
Philippa (nice way to honor both the DoE and Kate's sister, and "Princess Philippa of Cambridge"/"Queen Philippa" has a nice ring to it)
Sophia (Sophia of Hanover is the ancestress to everyone in the line of succession, so why not?)
Alexandra (obvious reasons)
Louise (traditional name)
Margaret (nice name, would honor William's great aunt)
Mary (not a favorite, though "Mary III" wouldn't be the worst)
Alice (a possibility, but not my favorite. Not modern sounding enough.)
I think there's much more room to go "non-traditional" (i.e. something not used by a British monarch) for obvious reasons, since there have been fewer female monarchs therefore fewer female names to choose from. I also don't think "Diana" will be used as a first name but I could see it as a middle name.
Clever. 
She's named after her paternal grandmother
