You're a friend of Tom's, aren't you?
Question: What does the 's in Tom's stand for?
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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
we don't normally say
*a friend of me
*a friend of Tom.
The idiomatic phrases are
a friend of mine
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.