Mark, there's a wide variety of, sometimes conflicting, advice on this thread; I'd really make sure you understand all of the implications of your next action before you commit yourself.
I think you would get good advice from a forum such as Pepipoo, in their Flame Pit sub-forum, (they are motoring specialists, we are for the most part engineers with other interests and opinions).
Make sure you are able to provide concise and unambiguous answers to the following questions in your first missive should you chose to post there.
How long ago was the vehicle bought by you?
Who currently holds the V5c and what is the DOCREF date on it? (the latest V5c is red; this replaced the previous blue one if you have two copies in your possession)
Who's name and address is on it? Is it the person who sold you the car?
Did the previous keeper actually own the car to sell? (i.e. is there HP on it, found by doing an HPI check?)
Who last taxed the car, how and when? (if taxed, by your son, prior to change of keeper, then the remaining tax will be cancelled, and returned to the previous registered keeper – not your son, when the vehicle is transferred to your son. He would then need to tax it again, or be in breach of legislation)
Your son will be the new registered keeper, there isn't a DVLA concept of registered owner.
N.B. all the insurance questions must be answered truthfully for obvious reasons.
As far as I'm aware, your son only has to provide an address where he can be reliably contacted. i.e. he could be the registered keeper, but at your address, though I'd check up on that in case the law has changes recently. N.B. this can also lead to further unintended complications, like council parking permits etc, as well as any NIPs for road traffic matters would be sent to your address, not his if he's flown the coop.
When I bought my van a few years ago, the seller and I changed me to registered keeper on-line at his premises; we then both kept printouts of the confirmation. I was then able to tax it on-line in my name.
Bill