Which one is correct?
- It is my friend's, Adam's birthday.
- It is my friend, Adam's birthday.
Does the word 'friend' need an apostrophe to show possession?
If using the nonrestrictive (as discussed in another answer), I would simply rephrase the sentence to avoid the possessive:
It's the birthday of my friend, Adam.
With this, there is no awkward question about where to put the possessive apostrophe. (Because it will look strange no matter where you put it.)
Your question is actually a perfect storm in which rules of puntuation dictated by grammar clash with the deeper grammar of spoken English, though rarely with the noun in your example.
Unless you are in the unfortunate situation of having only one single solitary friend in the world, in your example the proper name Adam is a restrictive appositive, that is, one necessary to restrict the meaning to one particular friend: your friend Monica as opposed to your friend Chandler. Restrictive appositives are never bracketed by commas, yielding:
my friend Adam’s birthday
A nonrestrictive appositive is absolutely identical to the noun to which it stands in apposition and thus requires bracketing commas:
His wife, Eleanor, is head of pediatrics.
Barring bigamy or polygamy, this guy has only one wife, so Eleanor and wife are specified to the same person. No one needs to ask which wife. Thus the commas.
It only gets weird if he wants to talk about his wife’s birthday and wants to include her name.
According to one source, to form the possessive of a noun and its appositive in written English, you treat the whole construction as a single noun phrase, making only the appositive possessive and omitting the second bracketing comma:
We need Mr. Smith, the family attorney’s signature.
Somewhere behind this sentence is a maniacal grammarian screaming about commas around nonrestrictive appositives but not knowing what to do with the second one.
With proper names, however, this construction can yield bizarre results:
I have to feed my brother, John’s dog.
She wrecked her best friend, Petra’s Volvo.
Now the comma is for the eye only — if proper names are nonrestrictive appositives, the commas are never marked by pauses. If someone says “my wife Rachel’s car,” it doesn’t mean there’s another wife Leah somewhere.
In these cases, I would simply omit the commas entirely, following the prosody of the spoken language rather than a punctuation rule that could misdirect a reader to suppose a canine sibling or a best friend who’s secretly a Swedish automobile.
Me? I go without any commas
It's my friend Adam's birthday
If I placed Adam in parenthesis, I'd write
It's my friend's (Adam) birthday.
Ignoring the apostrophes for the moment you either need to lose the comma before Adam or add another comma after it.