"Are you a friend of Rachel's?" is correct. It's what we call a double genitive. Look at this: a friend of mine a friend of yours a friend of his a friend of hers a friend of ours a friend of theirs. As you can see, in each case we use the possessive pronoun after 'a .. of ...'. We don't say 'a friend of me' or a 'friend of him', for example, and we don't usually say 'a friend of Rachel': we use the genitive (possessive) form of the name. "Are you Rachel's friend?" is also correct, but it has a slightly different meaning. We would say this when we are referring to a specific friend who has already been mentioned. For example, let's say Rachel had sent a friend of hers to call at your house. When a stranger knocks on your door, you'd then say "Are you Rachel's friend?". This means 'the particular friend that Rachel told me about'.

The second sentence is correct (Are you Rachel’s friend). It is also acceptable to ask, Are you a friend of Rachel’s? The apostrophe must be used in both sentences because the friend belongs to Rachel. It indicates possession. Other examples are: John’s dog is sick. Sally’s brother lives next door. Robin’s smile makes me happy.