From what you describe I think it quite likely that the debt is still owned by the original company, & they have simply moved it from one debt collector to another. This is quite a common action - particularly if the first debt collector has failed to get a response.
However, in most such cases the debt owner refuses to deal with the debtor - they insist that any negotiations & payments are made through the debt collector who has the debt at the time.
There is nothing to stop you making a "full & final settlement" offer. But if you do, make sure it is in writing & keep a copy. Include the words "full & final settlement" in your letter. Make it clear you are NOT the debtor, but his father. Also explain your son's present financial & family situation, making it clear he has no money due to unemployment. Make sure you get written confirmation that the offer is accepted in full & final settlement before you make a payment.
It is quite possible that a 50% offer will be accepted, but you must not assume this to be the case - the policies of debt owners & collectors vary quite a lot. I have seen offers much lower accepted, & some much higher rejected. Also, it quite often happens that the offer you make is rejected but if it is then queried a slightly higher offer is accepted, so allow a little leeway in your first offer so you can increase it a bit if necessary.