My father, Emanuel John Klironomas, immigrated to the USA in 1906 from the island of Crete, Greece.

From previous discussions with others I have been told that most surnames from the island of Crete, Greece ended in 'IKIS'

Does anyone know where the 'omas' ending came from.

Is it possible that your family name was originally Klironomos rather than Klironomas?
This is a Greek surname and in that era, upon immigration to the US, many immigrant's names were misspelled in both minor and major ways and many chose to simply accept the new spelling because fixing it was a truly daunting task, particularly for someone who had a limited command of spoken and written English.
In the case of a Greek immigrant, it's possible that a mistake went unnoticed due to the difference in the alphabets used by your father and the government clerk registering his arrival.... might be worth looking into.