What's in a name?
I don't think it makes the slightest difference what the place is called, and I doubt anyone at the college would be inclined to value the name over the donation. It's hardly a first anyway; Harris Manchester in Oxford was originally "Manchester" until Lord Harris gave it lots of money. Templeton College in Oxford was the Oxford Centre for Management Studies until John Templeton stumped up some cash. This sort of thing is quite normal, and no-one thinks of those colleges as somehow selling out or betraying their souls. It happened in the Middle Ages and the Renaissance too - Clare College in Cambridge was originally University Hall until Elizabeth de Clare bailed it out in the fourteenth century. Gonville and Caius was originally Gonville Hall. And so on. And of course, colleges are quite capable of using nicknames for so long that the "official" names fall out of use (Oriel in Oxford is an example), which shows that names have never been regarded as particularly sacrosanct.
Also, where's this "Oxbridge"? New Hall is in Cambridge!
All of the college you mention were "new" when they changed their names. Who cares what the "Oxford Centre for Management Studies" is called? It has minimal or no historical assocations with its name.
This is not true of New College.
Apples and oranges, or at least apples and pears.
Gonville was over two hundred years old when Caius came along. New Hall is only fifty years old. (You might be mixing it up with New College, Oxford?) But even if it weren't, I'm not convinced that even a terribly old name has much value. The colleges at Oxford and Cambridge have always had a very pragmatic attitude to most things, and have generally preferred survivability to tradition. That's how they've survived for so long.