That would be pretty cool to designate a principality (Princedom) as the traditional province of prince/princesses which who is the heir apparent. I can see at as a kingdom choice (also one of the perks of having a kingdom/empire), which lets you *pick* the county or duchy for this.
A principality the size of county would give the prince the title of "Prince of [County Name]". For example, the Prince of Kent, or the Prince of Leon.
A principality the size of a duchy would give the prince the title of "Prince of [Duchy Name]". For example, the Prince of York, or the Prince of Anjou.
Once you set the principality, you can change it after the fact, but it should be pretty costly so you don't see people flipping the principality from place to place. Perhaps, the larger the principality, the better the bonuses, but also in some cases could cause large problems - such as a prince who is rebelling against the father - and because of his vast principality it would be hard to crush the rebellion. Also, larger principalities could cause stress and other health issues due to the increased responsibilities.