I was recently reading the books, and I was curious to know what sort of real world basis might have existed in the Medieval period. We have Vice Presidents, and so on, but what did rulers used to rely on?
Would not the Grand Vizier of the Ottaman Sultanate be alike to the Hand of the King?
There were powerful ministers and in Spain for instance, although later the figure of the "Valido" existed or a kind of all powerful minister who de factor controlled the kingdom. You can also see other figures in absolutist France like Cardinal Richelieu being both a top Church figure and a kind of de facto ruler deciding about war, foreign policy etc.
There was a Protector of the Realm who took charge when the king was unable to. Henry V (I think, it could be Henry IV but pretty sure it was Henry V) was mentally ill for a few years and so Richard of York had to rule in his stead, which Henry's wife wasn't happy about. So in the way Ned plans to rule before Jeffrey is of age, a Protector of the Realm in England would rule until the heir was old enough. This position could also be taken by regent, such as Cersei does.