My guess is [name]Eleanor[/name] for a girl- [name]Eleanor[/name] of [name]Aquitaine[/name] was queen 850 years ago, so her story seems distant enough not to be a burden to a modern member of the royal family (not that it would necessarily be a burden!). [name]Queen[/name] [name]Eleanor[/name] just plain sounds right, plus itās the sort of name that would give her family the option of a private nickname.
We had a couple of [name]Queen[/name] [name]Marys[/name] at the beginning of the modern age, and Iām not sure that Iād want to be [name]Queen[/name] [name]Mary[/name] III when the first [name]Mary[/name] is still known as āBloody [name]Mary[/name]ā and the second one is somewhat of a traitorous character to the Scots.
[name]Alice[/name] is another possibility, I think, given that it is a [name]Royal[/name] family name currently in fashion, and it has the benefit of not having an obvious historical precedent. On the other hand, [name]Alice[/name] is almost a trendy name, and Iām not entirely convinced that ā[name]Queen[/name] [name]Alice[/name]ā quite stands upā¦Other more remote possibilities could include [name]Margaret[/name] (though that could be tricky in [name]Scotland[/name]- [name]Margaret[/name] I in [name]England[/name], II in [name]Scotland[/name]?), [name]Alexandra[/name] (royal precedent, excellent etymology for a queen, but a little much- [name]Princess[/name] [name]Alexandra[/name], [name]Queen[/name] [name]Alexandra[/name]? Too much [name]Victoriana[/name]!), [name]Charlotte[/name], [name]Helena[/name], and, remotely, [name]Matilda[/name] or [name]Emma[/name].
I agree with [name]Elizabeth[/name] [name]Diana[/name] [name]Caroline[/name] as obvious middles, but some combination including [name]Alexandra[/name], [name]Margaret[/name], [name]Frances[/name], [name]Alice[/name], [name]Mary[/name], [name]Anne[/name], or [name]Catherine[/name] also make a lot of sense. Best guess is [name]Eleanor[/name] [name]Diana[/name] [name]Elizabeth[/name] [name]Caroline[/name]; runner up is [name]Charlotte[/name] [name]Frances[/name] [name]Alexandra[/name] [name]Mary[/name].
Boys are interesting- do they stick with one thatās been done- [name]George[/name] VII? [name]Charles[/name] IV? [name]Edward[/name] or [name]Henry[/name] IX? [name]One[/name] with a much more major numbering issue (never mind sticky historical connotations) - [name]James[/name] III/VIII? [name]Stephen[/name], [name]John[/name], and [name]Richard[/name] seem unlikely because of the popular perception of the last monarchs with those names. I think [name]George[/name] and [name]Edward[/name] are the best bets from the basic list of already-done English kings, but a seventh or ninth is more than a little dull to contemplate (granted the āninthā part is unprecedented for the Commonwealth crown).
If they do decide to go out on a limb and pick a ānewā name, clearly it would have to be a family name and/or one with deep, deep British roots. [name]Arthur[/name], [name]Philip[/name], [name]Michael[/name], [name]Alexander[/name], [name]Frederick[/name], [name]Albert[/name], and [name]David[/name] are the obvious āfamilyā names. [name]Arthur[/name] is just wayyy too much to put on a person who will be [name]King[/name] of [name]England[/name]. [name]David[/name], [name]Louis[/name]/[name]Lewis[/name], and [name]Alexander[/name] are eliminated for reasons similar to [name]Arthur[/name]; [name]Michael[/name] and [name]Philip[/name] are sensible middle names, but somehow I cannot imagine them as the first name of a child born to that couple; [name]Frederick[/name] and [name]Albert[/name] I think are a bit too [name]German[/name]/Prussian, especially when you start sticking ā[name]Prince[/name]ā and ā[name]King[/name]ā in front of them-- though [name]Albert[/name] is the best of the lot. There are a couple of very old names that I think could potentially work for a late 21st-century British king- [name]Edmund[/name] and perhaps [name]Alfred[/name]. There are a handful of pre-1066 [name]Edmunds[/name], but long enough ago that they are arguably more myth than history, if known at all, for most. The more immediate association of course is the [name]Edmund[/name] who is a ākingā of [name]Narnia[/name]āwhich, I think, works. [name]Edmund[/name] is certainly the most interesting of the four children, and modern [name]Britain[/name] certainly takes pride in its literary tradition (hello, Olympic opening and closing ceremoniesā¦). [name]King[/name] [name]Alfred[/name] is of course the (semi-legendary, the Victorians had a bit of a love affair with him) king of the Anglo-Saxons who is mostly credited with establishing education as the basis for a successful kingdom/economy, among other things. There are worse precedents, anyways 
Best guess hereā¦going out on a limb with [name]Edmund[/name] [name]Arthur[/name] [name]Philip[/name] [name]Michael[/name]; runner-up safe bet would be [name]George[/name] [name]Alexander[/name] [name]William[/name] [name]James[/name].