Monyo (blood line, lineage, connected by blood) (門葉)
The term "monyo" generally referred to families that were related by blood. Later, it was also used for the rank name in reference to a social status among vassals who were related by similar blood.
Gomonyo (a family with honorary status)
The term "gomonyo" referred to those who, among the Seiwa-Genji (Minamoto clan), were treated as a family of Kamakura-dono (Lord of Kamakura) in the Kamakura bakufu (Japanese feudal government headed by a shogun). In the period of three generations of the Genji shogunate family, from MINAMOTO no Yoritomo to MINAMOTO no Sanetomo, the Genji clans that belonged to the same clan of Kamakura-dono existed in the Kawachi-Genji (Minamoto clan) in a restricted sense and in Seiwa-Genji in a broad sense, but not all Genji clans were always treated as the family of Kamakura-dono because the loyalty to Kamakura-dono was highly valued and dependence on the family was not permitted.
Many Genji clans, except for the conferment of a court rank and appointment to an office, were forbidden to broadly share a common Minamoto family name in the same way as the Kamakura-dono, and were treated simply as Kenin (retainers).
When MINAMOTO no Yoritomo established the military government in Kamakura, Yoritomo allowed those who had been credited for their distinguished services among the Seiwa-Genji the permission to use the Minamoto family name, but he prohibited other Genji clans from using the name at formal occasions. Monyo originated from six people who were appointed as Kokushi (provincial governors) at Jimoku (ceremony for the appointment of officials) on August 16, 1185 (old lunar calendar). Families that were acknowledged as monyo were called gomonyo and treated second to the shogun family in social status; this drew the line at other gokenin (an immediate vassal of the shogunate in the Kamakura and Muromachi through Edo periods). Therefore, it was considered a great honor to be acknowledged as a gomonyo.
In fact, the first six people had been recommended before Yoritomo got onto bad terms with MINAMOTO no Yoshitsune. Also, the relationship between Yoritomo and Yoshitsune was strained at the point that Jimoku was made public, so Yoshitsune wasn't factually acknowledged as a gomonyo. Given the manner in which the members of these monyo were selected, in order to be given the status of monyo, what mattered wasn't the main line or branch of the family blood nor how close one was to Yoritomo's blood but was instead being the faithful Seiwa-Genji, which was the only requirement.
Yoritomo's own younger brothers, MINAMOTO no Noriyori, Zenjo ANO and Yoshitsune, weren't included in these monyo, while those included were limited to Yoshinori YAMANA, a child born out of wedlock of the Nitta clan, among the Kawachi-Genji that had relatively close blood ties with Yoritomo, and Yoshikane ASHIKAGA; however, those who were distantly related were also included, such as Tomitsu KAGAMI of the Kai-Genji that had descended from MINAMOTO no Yorimitsu, the younger brother of MINAMOTO no Yoshiie, the ancestor of four generations before Yoritomo, and Yoshisuke YASUDA; and MINAMOTO no Aritsuna of the Settsu-Genji (Minamoto clan), which was descended from MINAMOTO no Yorimitsu, the elder brother of MINAMOTO no Yorinobu, the originator of the Kawachi-Genji, the ancestor of six generations before.