Why use apostrophe-s to denote possession when using 'of'

Think of the simple phrase "Bill's friend".

If you were going to turn this around using the preposition 'of' would you say:

• A friend of Bill's or
• A friend of Bill

It appears to me that, in the US anyway, people always say "A friend of Bill's".

Even though I'm a native American English speaker, this just sounds weird to me. It seems to create a 'double possessive' (a term I just invented). I always want to respond: "A friend of Bill's what?" A friend of Bill's aunt?

So what's at work here, and do British English speakers also do this?

Solution 1:

As @FX_ points out, it’s called a double genitive or double possessive.

In this example, it’s not compulsory: both a friend of Bill’s and a friend of Bill are correct, although the first is probably more common. (Usage data, anyone?)

If Bill were replaced by a pronoun, however (poor Bill!), the double genitive would be required: a friend of mine is correct, but not a friend of me. (Similarly with yours vs. you, his vs. he, etc.)

Also, sometimes, this is needed to avoid ambiguity between the possessive and other uses of of: for instance, a picture of Bill’s means that he owns the picture, whereas a picture of Bill means he’s portrayed in it.

Solution 2:

The correct form is “a friend of Bill’s”, and it's called a double genitive (see also here).