In medieval Europe, "son of the king" would not have been an office with any power. (Princeps, or prince, can be used for any ranking noble in Latin sources--including, in the Holy Roman Empire, the emperor himself). However, sons of kings, including the heir apparent, were granted titles of their own that would convey authority. The crown prince of Britain today is the Prince of Wales, then his son is the Duke of Cambridge, and so forth.

In the later medieval HRE, whose nobles technically elected the emperor (and yet the position was mostly hereditary), the probable heir would be crowned King of the Romans. There are even a few leaders of the HRE who never bothered to secure formal coronation as emperor, and were technically just King of the Romans for their whole reign.