Claudius was given the agnomen Germanicus before his accesion, and retained it no doubt because of his brother when he became emperor. When offered the title Britannicus after his conquest of Britain, he refused it but gave it to his son, who was from then on known as Britannicus.
So basically the agnomen is strictly personal and based on military feats, the only thing missing is why Claudius was given the agnomen Germanicus in the first place.
Interestingly, the agnomen Germanicus was originally also bestowed on the father, Nero Claudius Drusus, who also accepted it on behalf of his son, 'Germanicus'.
edit: The title Germanicus does seem to have been hereditary. From wiki: His (Drusus') family was granted the hereditary honorific "Germanicus", which was given to his eldest son before passing to his youngest.