The expression "one of the ..." or "one of my ..." etc always takes a noun in the plural -- One of the best cars in the world ... When you write "One of my friends' friends" the first word "friends'" is a possessive modifying the head noun, which is the second word "friends". The head noun must be plural after "One of the ...".

I assume that your friend has more than one friend, too, so the first answer ("one of my friends' friends") is correct.