You're a friend of Tom's, aren't you?
Question: What does the 's in Tom's stand for?
You're a friend of Tom's, aren't you?
Question: What does the 's in Tom's stand for?
It doesn't stand for anything: it is simply the possessive clitic exactly as in Tom's friend.
There is an unobvious rule in English that though we say
my friend
and
Tom's friend
we don't normally say
*a friend of me
or
*a friend of Tom.
The idiomatic phrases are
a friend of mine
and
a friend of Tom's".
I hesitate to say the other forms are ungrammatical, but they are certainly not natural in current English. I think I can imagine a converation like
Who is that man over there? I don't know, I thought he was with you. I think he must be a friend of Tom.
Even in that context, "a friend of Tom's" is more likely, but I think "of Tom" is possible.
It's another way of saying "You're one of Tom's friends, aren't you?"
The 's is the possessive.