I found out that 古希 / こき can mean a 70 year old person, or the occasion of turning 70 years old. Apparently its etymology derives from an ancient song where this line occurs:七十古希 【しちじゅうこき】 which means "Men seldom live to be seventy" (Du Fu (c.712-c.770)); (Few people live to be seventy). Was 古希 a chinese word meaning rare or uncommon? Is it used mainly to describe the special occasion of someone's seventieth birthday or is it used to describe one's seventieth year in the same way as はたち?
古希, also 古稀. 希 appears to simply be a simplified version of 稀. The word actually comes from a poem, rather than a song.
古稀 technically means 70 years of age, but could sometimes be interpreted as 70th birthday, as in 古稀の祝い, "70th birthday celebration".
Here are the Chinese line 古稀 derives from, a Japanese transcription thereof, and an English translation thereof.
Firstly, you may note that the Chinese line does not actually say "古希" as you suppose in your question. Secondly, it becomes apparent that 古来稀なり is the origin of 古稀, with 来 and なり simply dropped at some point. But it basically means the same thing..."(that which) has always been rare", i.e., living to be (or being) 70 years old.
And if the poem in full should interest you...
By way of a random blog, a decent translation of the poem:
For much more detail - in Japanese - you can check what a site called A Rainy Day (and its section on idioms derived from Chinese literature/history) has to say about 古稀. Specifically on that page, under 現代語訳 is a more complete explanation of the poem.
The basic idea is: you rack up debts for wine, yet life is short (living to 70 is rare)...why don't we take the time to watch the butterflies, the dragonflies, et cetera, and simply enjoy ourselves for a while?