Onzoshi' in history
This refers to a son who was not the oldest in the samurai class families in the Middle Ages in Japan. That is to say, he was supposed to be respected as the brother of the future head of the family, but was not necessarily a person the feudatories were loyal to, as he would not inherit his father's property.
The term was used from the end of the Heian Period to the beginning of modern times, and in the Edo Period the son was called heyazumi (literally, a person living in a room)
This 'Onzoshi' mainly referred to the sons of the Minamoto clan such as MINAMOTO no Yoritomo and MINAMOTO no Yoshitsune. On the other hand, the sons of the Taira clan, which was also a samurai family but rapidly changed into court nobles in TAIRA no Kiyomori's times, were called 'Kindachi'.
Onzoshi' in modern terms
Today, 'Onzoshi' mainly has two meanings.
One meaning is a son who is born into a Kabuki family and supposed to succeed to one of the family's stage names when he becomes an adult. For example, Ebizo ICHIKAWA (the eleventh) and Kikunosuke ONOE (the fifth) are onzoshi of the Naritaya family and the Otowaya family respectively.
The other meaning is a son who is born into a wealthy family and is eventually expected to inherit its property. In this case, it has a nuance of 'o-bochan' (a son who is well brought up) in a good meaning, and 'dora-musuko' (a prodigal son) in a bad meaning.