This one like so many other words in our vocabulary seem to be shortened down from a lengthy phrase. I am not 100% sure this is the true meaning, but hey, it sounds good, and I found it in a couple sources.

It seems to be an old term for a longevity pay increase. Believed derived from the paymaster's log entry: 'For On Going Years', or the initials: F.O.G.Y. It has become an insulting term for anyone with old fashioned notions or beliefs.

Originally it did not mean an old person, but a person long in service. Many times young boys, as young as 9, would begin their career with a Navy and spend the next 50 or more years in service. So a young man of 39 or 40 could be a FOGY.