//If the son is the registered keeper, would the notice not be sent to his address to ask who was driving it? //
Yes Corby. That's where I'm coming from. There is a possibility that the OP's son has been subject to criminal proceedings of which he was not aware and there's ways of dealing with that.
But more than that, all we know is that "the notice" (what notice?) was sent to his old partner's address. We're also told that "Because he’s now notified the DVLA of his new address, the fine has increased, because he failed to pay on time.". That doesn't happen with speeding fines.
So, malagabob, please tell us (1) was your son stopped at the time or was he caught by camera? (2) Exactly what has your son received, who is it from and what does it say? (3) If he was not stopped at the time, what correspondence was exchanged which would provide the evidence that he was driving?