Moretti (1957), no. 645 suggests Gnaeus was the name of Menodoros' father. [DvdL]
The man prides himself on having won the periodos, but there are no Isthmian victories on either of his two victory lists, wherefore Farrington considers him a 'doubtful periodonikes', and Moretti even calls him 'un falso periodonikes'. However, the problem may be solved if the Isthmia were in abeyance at that time, or if there periodos had not been strictly defined yet, see Farrington (2012), n. 687.
The Delian inscription ends with the δῆμος of Athens, Rhodes, and Boiotian Thebes and king Ariarathes V (a Cappadocian king) honouring Menodoros. It is not clear whether Menodoros held honourary citizenship at these places. The mention of Ariarathes V who died in 130 BC provides a terminus ante quem for this inscription.
NB: Notwithstanding the different states of preservation of the inscriptions from Delos and Athens, they do not seem to be exact copies of Menodoros' victor list.