About the possessive case ('s)

I would choose the second as the correct one.
"My father's cousin works at Macy's in New York".


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Since in:
"A cousin of my father's works at the Macy, in New York"

A cousin of --- it´s already a possessive, there´s no use a double possessive in English. [In cases like the current one.]



When you say "my father´s" somebody expects that you state what is "of your father".
The second apostrophe makes the sentence confusing, you could say (in another situation) my father´s work (the work of my father). as in:'Did you not know that I must be about my Father`s work? (Biblical passage of the finding of Jesus in the temple.) or in "“I can remember the stories of my father's work as the Vice President of Development at Covenant College."


But not here, since you have spend your "possessive quota" with "A cousin of my father..."
It makes better sense if you say "A cousin of my father works..." (that is, a cousin works - and - a this cousin is a cousin of my father. Not mine or of everybody else.)

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Back to the initial sentence I have chosen.

"My father's cousin works at Macy's in New York". is equivalent to

1) A cousin of my father works at Macy´s in New York. (Macy´s a commercial [brand] name, so having nothing to do apostrophe indicating possessive or anything) but...


2) Why they, have chosen Macy´s, in the first place?
Chances are that they meant Macy´s (Macy´s store), since apostrophes can be used to mean that (one´s office/house etc, implicit). That lead us to the notice #3.
http://www1.macys.com/shop/store?cm_sp= ... stores-_-n



I going to dentist´s (dentist´s office)
I will go to friend´s for dinner. (friend´s house)


3) A cousin of my father works at Macy´s in New York.
A cousin of my father works at Macy store in New York.



#3 being the corollary of #2.


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Just for information´s sake.
http://linguapress.com/grammar/possessives.htm