Could you tell me which is correct to say,
"Today is one of my friends' birthday." or "Today is my friend's birthday."
Thank you for your advice!
This depends on your intended meaning, as the examples convey different meanings.
Friends is the plural form of friend.
See here for a dictionary definition.
In the first example you cite, the meaning is you have more than one friend and today one of them has a birthday.
In the second example you are saying that your friend has a birthday (but not necessarily that you have more than one friend).
You sound somewhat more popular when you go with the first as you are conveying two pieces of information and not just one. (1) you have more than one friend, (2) one of them has a birthday today.
However, "Today is my friend's birthday" reads less awkward and is used more frequently idiomatically.