I would be surprised if it were Oliver given Oliver Cromwell had Charles the first beheaded. Maybe enough time has passed though!
Eugenie won't be restrained like Kate and William were but I think she will go reasonably traditional, I think names like Arthur, David, Nicholas, Hugh, Thomas, Frederick are good shouts. I think Philip is likely to be in there as a middle name but a good first name option too - it's a classic name which isn't particularly popular at the moment and it honours the DoE.
I don't think they will go as modern as Arlo for example but a name like Oscar might work.