It really doesn't mean rejoice. It's an alternative spelling of 'Gaius', which was the most common male first name in Rome (the C/G sound is interchangable in early Latin). The origin isn't clear.
The Cambridge way is a pretty specific pronunciation and certainly wouldn't occur to most people. Caius would correct everyone, of course, but 'Cai-us' is a nicer pronunciation, in my opinion. The college was named after John Caius, who was born John Kays (Caius was the Latinised version). Thus, 'Keys' isn't the proper way to say the name anyway.
Having said all that, I'd love to use Gaius/Caius - it's an awesome name. Just, if you want to call him Keys, call him Keys...