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

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.

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