The word γενέθλια comes from the ancient Greek verb γίγνομαι (in Past Tense εγενόμην), which means "I am born" (when it refers to people.)
The root γεν- with the ending -θλον gave the word γένεθλον (ancient Greek)= day of birth
The word γενέθλια is the neuter plural form of the adjective γενέθλιος, γενέθλια, γενέθλιον, which comes from the noun γένεθλον.
In modern Greek the word γενέθλια is used as a noun.
It is neuter plural (τα γενέθλια) by analogy with other celebrations whose name is neuter plural (π. χ. Χριστούγεννα = christmas)