Interesting question. The line of Peter's successors are well-documented. Barnabas is mentioned on this wikipedia page that refers to several lines have also been recorded.
The Patriarchate of Constantinople claims unbroken succession to the Throne of Saint Andrew.
The Greek Orthodox Church of Alexandria claims unbroken succession to the Throne of Saint Mark[18]
The Russian Orthodox Church claims unbroken succession to the Throne of Saint Andrew[19]
The Armenian Apostolic Church claims unbroken succession to the Thrones of Saint Bartholomew and Saint Jude Thaddeus[20]
The Coptic Orthodox Church of Alexandria claims unbroken succession to the Throne of Saint Mark[21]
The Assyrian Church of the East, including the Saint Thomas Church (e.g., Indian (Malankara) Orthodox Church[22]) claims unbroken succession to the Throne of Saint Thomas
The Orthodox Church of Cyprus claims unbroken succession to the Throne of Saint Barnabas[23]
The Ethiopian Orthodox Church claims succession to the Throne of Saint Philip[24]
The Greek Orthodox Patriarch of Jerusalem claims succession to the Throne of Saint James the Just,[25] although this line includes Patriarchs in exile.[26] (see Orthodox Patriarch of Jerusalem )
The Roman Papacy of the Catholic Church claims unbroken succession to the Throne of Saint Peter called "Prince of the Apostles" (see Petrine Succession )
*claims*.
Isn’t that nice.....