Hello,
"A friend of" is a possessive form, so to add the "-'s" to the person is double. It is correct to say "A friend of my dad", but in California we often say "A friend of my dad's" anyway. It is just sloppy English that has become acceptable. I don't know how people in other parts of the country speak, but sometimes it can be quite different.
In the same way that "A friend of mine" is correct whilst "A friend of me" is clearly not, "A friend of John's" and "A friend of my father's" are correct, even if they can sound more awkward than the less 'correct' forms, "A friend of John / my father". Of course, as has been pointed out, it is sometimes better to recast such phrases - "One of my father's friends".
That this 'double genitive' sometimes 'feels' illogical does not make it incorrect, and certainly does not make it sloppy English, - quite the contrary - though it can sometimes feel awkward : I guess you'd just have to go with what feels 'right' to you.
This usage always 'feels' right with pronouns, doesn't it ? - "That old car of OURS needs a good wash" (try 'US' !) - and more often than not 'feels' right with singular nouns and names "That dirty old bone of the dog's wants throwing in the bin" (less happy, perhaps, is 'dog'), or "It's about time we got rid of those old toys of Peter's" ('those old toys of Peter' !?)
Where the wheels do start to come off is when using plural forms, or double nouns - "It's about time we got rid of those old toys of Peter's and Mary's / of Peter and Mary's", "It's about time we got rid of those old toys of the children's" : better maybe to say instead, "It's about time we got rid of the children's old toys".
Of course, there are usages which don't take this 'double-genitive' and which would be awkward with it - 'The Marriage of Peter and Mary will take place &c.', 'The funeral of Mickey Mouse was held at the Rodents Retreat' (Rodent's / 'Rodents' ... ? Pace Lynne Truss' book 'Eats, Shoots & Leaves', let's please not go there !') in other words, do the events 'belong' to Mary & Peter or to the late Mr Mouse, in any meaningful sense ... ?
Usage is a quagmire, for while "The Marriage of Figaro" is undoubtedly correct, so also is the title of Browning's poem, "A Toccata of Galuppi's", a title which puzzled me much as a schoolchild until I cottoned on to the 'double-genitive'.
(The non-'double-genitive' - or whatever it's called - will doubtless gain ground. I am told it is all a matter of word order : expecting to see and hear gentives before nouns, rather than well after them, we adjust accordingly and say "A friend of my Dad" rather than "A friend of my Dad's". The aforementioned Lynne Truss writes of this in that funny-old book of hers.
We see the same thing happening with possessive clauses - "John is a bore, and HIS / HIM turning up out of the blue really put the kybosh on the party." Whilst 'his' is correct, a genitive is mistrusted before a noun-phrase and the object form 'him' is often preferred.
Again, in the phrase 'Between you and ME / I' - the subject form 'I' is often preferred because it feels 'right' at the start of a sentence, though the object form 'me' is plainly indicated.
And Jesus' "Let him who is without sin ... " is often incorrectly rendered as "Let HE who is without sin ... ", because the subject form 'he' seems correct at the head of the phrase ' ... he who Is without sin', whereas the object form 'him' is required after the verb 'Let'. And if you don't believe anyone would write this, just google it and see for yourself ! ... And there's always the title of the Startrek episode['s], isn't there ?!)
Ian